The Differential Equations for the Simplest Types of Vibrations
On several occasions, we have already encountered differential equations, i.e., equations from which an unknown function is to be determined and which involve not only this function but also its derivatives.
The simplest problem of this type is that of finding the indefinite integral of a given function f(x). This problem requires us to find a function y = F(x) which satisfies the differential equation y'-f(x)=0. Moreover, we solved a problem of the same type in 3.7, where we showed that an equation of the form y'=ay is satisfied by an exponential function y = cea x . As we saw in 5.4, differential equations arise in connection with the problems of mechanics; indeed, many branches of pure mathematics and most of applied mathematics depend on differential equations. In this chapter, without going into the general theory, we shall consider the differential equations of the simplest types of vibrations. These are not only of theoretical value, but also extremely important in applied mathematics.
It will be convenient to bear the following general ideas and definitions in mind. By a solution of a differential equation, we mean a function which, when substituted into the differential equation, satisfies the equation for all values of the independent variable under consideration. Instead of solution, the term integral is often used: In the first place, because the problem is more or less a generalization of the ordinary problem of integration, and, in the second place, because it frequently happens that the solution is actually found by integration.
11.1 Vibration Problems of Mechanics and Physics
11.1.1 The Simplest Mechanical
Vibrations:. The simplest type
of mechanical vibration has already been considered in 5.4.3. We have considered
there a particle of mass m which is free to move on the x-axis
and which is brought back to its initial position x = 0
by a restoring force. We assumed the magnitude of
this restoring force to be proportional to the displacement x;
in fact, we equated it to -kx, where k is a
positive constant and the negative sign expresses the fact that
the force is always directed towards the origin. We shall now
assume that there is also a frictional force present which is proportional to the velocity 
of the
particle and opposed to it. This force is then given by an
expression of the form 
, with a positive frictional
constant r. Finally,
we shall assume that the particle is also acted on by an external
force which is a function f(t) of the time t.
Then, by Newton's fundamental law, the product of the mass m and the
acceleration 
must be
equal to the total force, that is, the elastic force plus the
frictional force plus the external force. This is expressed by
the equation
This equation determines the motion of the
particle. Recalling the previous examples of differential
equations, such as the integration problem 
with its solution 
or the
solution of the particular differential equation 
in 5.4.3, we observe that these
problems have an infinite number of different solutions. Here
too. we shall find that there are an infinite number of
solutions, which are expressed in the following way. It is
possible to find a general solution or complete
integral x(t)
of the differential equation, depending not only on the
independent variable t, but also on two parameters c1
and c2, called the constants
of integration. If we assign
special values to these constants, we obtain a particular
solution, and every solution
can be found by assigning special values to these constants. The
complete integral is then the totality of all particular
solutions.
This fact is quite understandable (5.4.4). We cannot expect that the differential equation alone
will determine the motion completely. On the contrary, it is
plausible that at a given instant, say, at the time t = 0, we
should be able to choose the initial position x(0) = x0
and the initial velocity 
(in short, the initial
state) arbitrarily; in other
words, at time t = 0, we should be able to start the
particle from any initial position with any velocity. This being
done, we may expect the rest of the motion to be definitely
determined. The two arbitrary constants c1
and c2 in the general solution are just
enough to enable us to select the particular solution which fits
these initial conditions. In 11.2.1, we shall see that this can be done in one way only.
If there is no external force present, i.e., if f(t) = 0, the motion is called a free motion. The differential equation is then said to be homogenous. If f(t) is not equal to zero for all values of t, we say that the motion is forced and that the differential equation is non-homogeneous. The term f(t) is also occasionally referred to as the perturbation term.
11.1.2 Electrical Oscillations: A mechanical system of the simple type described can
actually be realized only approximately. An approximation is
offered by the pendulum, provided its oscillations are small. The
oscillations of a magnetic needle, the oscillations of the centre of a telephone
or 
microphone diaphragm and other mechanical vibrations can be
represented to within a certain degree of accuracy by systems
such as we have described. But there is another type of
phenomenon which corresponds far more exactly to our differential
equation. This is the oscillatory
electrical circuit.
We will consider the circuit sketched in Fig. 1
with inductance m, resistance r and capacity C = 1/k. We also assume
that the circuit is acted upon by an external electromotive force j (t) which is known as a function of the time t,
such as the voltage supplied by a dynamo or the voltage due to
electric waves. In order to describe the process taking place in
the circuit, we denote the voltage across the condenser
by E and the charge in the condenser by Q. These quantities are
then connected by the equation CE = E/k = Q.
The current I, which like the voltage E is a
function of the time, is defined as the rate of change of the
charge per unit time, i.e., as the rate at which the charge on
the condenser drops: 
Ohm's law states that the product of the current and the
resistance is equal to the electromotive force (voltage), i.e.,
it is equal to the condenser voltage E minus the counter
electromotive force due to self-induction plus the external
electromotive force f(t). We thus arrive at the equation
or
which is satisfied by the voltage in the circuit. Hence, we see that we have obtained a differential equation of exactly the type considered in above in 10.1.1. Instead of the mass, we have the inductance, instead of the frictional force the resistance and instead of the elastic constant the reciprocal of the capacity, while the external electromotive force (apart from a constant factor) corresponds to the external force. If the electromotive force is zero, the differential equation is homogeneous.
If we multiply both sides of the differential equation by -1/k and differentiate with respect to the time, we obtain for the current I the equation
which differs from the equation for the voltage only on the right hand side and has for free oscillations (f = 0) the same form.
11.2 Solution of the Homogeneous Equation. Free Oscillations
11.2.1 The Formal Solution:
We easily obtain a solution of the homogeneous equation 
in 11.1.1 in the form of an exponential expression by seeking to
determine a constant l in such a way that the expression elt
= x is a solution, If we substitute this expression and
its derivatives 
' in the differential equation and remove the common
factor elt, we obtain the quadratic equation
for l. The roots of this equation are
Each of the two expressions 
is, at least formally,
a particular solution of the differential equation, as we see by
carrying out the calculations in the reverse order. There can now
occur three different cases:
1. r²
- 4mk > 0: The two rootsl1 and l2 are then real, negative and unequal; we have two
solutions 
With the help of these two solutions, we can at once
construct a solution with two arbitrary constants. In fact, on
differentiation, we see that
is also a solution of the differential equation. We shall show in 10.2.3 that this expression is in fact the most general solution of the equation, i.e., that we can obtain every solution of the equation by substituting suitable numerical values for c1 and c2.
2. r²
- 4mk = 0: The quadratic equation has a double root. Thus, to
start with, we have,apart from a constant factor, only the single
solution 
But we readily verify that in this case the function
is also a solution of the differential equation. In fact, we find that
and substituting we see that the differential equation
is satisfied. Then the expression
yields a solution of the differential equation with two arbitrary integration constants of c1 and c2.
We are led to this solution naturally by the
following limiting process:If l1¹l2, then the expression 
also represents a
solution. If we now let l1 tend to l2 and write l instead l1, l2,
our expression becomes 
3. r²
- 4mk < 0: We set 
and obtain two solutions of the differential
equation in complex form, given by 
Euler's formula 
yields for
the real and imaginary parts of the complex solution u1,
on the one hand, 
and,
on the other hand,
We see from the second version that v1
and v2 are (real) solutions of the
differential equation. We verify this directly by differentiation
and substitution.
We can again form from the two particular solutions a general solution
with two arbitrary constants c1 and c2 . This result may also be written in the form
where we have set c1 = a cosnd, c2 = a sinnd, and a, d are two new constants.We recall that we have already come across this relation for the special case r=0 in 5.4.3.
11.2.2 Physical Interpretation
of the Solution: In the two cases 
, the solution is given by the
exponential curve or by the graph of the function 
which for
large values of t resembles the exponential curve, or by
the superposition of such curves. In these cases, the process is
aperiodic, i.e., as the time increases, the distance x approaches
the value 0 asymptotically, without oscillating about the value x
= 0. Hence, the motion is not oscillatory. The effect of friction
or damping is so large that it prevents the elastic force from
setting up oscillatory motions.
It is quite different in the case 
, where
the damping is so small that one has complex roots l1, l2.
The expression 
now yields damped harmonic oscillations. These are oscillations which follow the sine law and
have the circular frequency
but the amplitude of which, instead of being
constant, is given by 
the i.e., the amplitude decreases
exponentially; the larger is r/2m, the faster
is the rate of decrease. In the physical literature, this damping
factor is frequently 
called the logarithmic
decrement
of the damped oscillation, the
term indicating that the logarithm of the amplitude decreases at
the rate r/2m. A damped oscillation of this kind is
shown in Fig. 2. As before, we call the quantity T = 2p/n the
period of the oscillation and the quantity nd the phase
displacement. For the special case r = 0, we obtain
again simple harmonic oscillations with the frequency 
, the natural
frequency of the undamped
oscillatory system.
11.2.3 Fulfilment of Given
Initial Conditions. Uniqueness of the Solution: We must still show that the solution with the two
constants c1 and c2 can
be made to fit any pre-assigned initial state, and also that it
represents all the possible solutions of the equation. Suppose
that we have to find a solution, which at time t = 0
satisfies the initial conditions 
where the numbers 
can have
any values. Then, in Case
1 in 11.2.1, we must set
Hence we have for the constants c1 and c2 have two linear equations with the unique solutions
In Case 2 of 11.2.1, the same process yields the two linear equations
from which c1 and c2 can again be determined uniquely. Finally, in Case 3 of 11.2.1, the equations determining the constants assume the form
with the solutions
Thus, we have shown that the general solution can be made to fit any initial conditions. We have still to show that there is no other solution. For this purpose, we need only show that, for a given initial state, there can never be two different solutions.
If two such solutions u(t) and v(t) were to exist, for which
then their difference w = u -
v would also be a solution of the differential equation
and we should have 
Hence, this solution would correspond to
an initial state of rest, i.e., to a state in which at time t
= 0 the particle is in its position of rest and has zero
velocity. We must show that it can never set itself into motion.
In order to do this, we multiply both sides of the differential
equation 
and recall that
We thus obtain
If we integrate between the times t = 0 and t = t and use the initial conditions
we find
However, this equation would yield a contradiction, if at any time t > 0 the function w differed from 0. In fact, the left hand side of the equation would be positive, since we have taken m, k and r to be positive, while the right hand side is zero. Hence w = u = v is always equal to 0, which proves that the solution is unique.
Find the general solution and
also the solution for which
6. Find the general solution, and also the
solution, for which 
of the equation
Determine the frequency (n), the period (T), the amplitude (a), and the phase (d) of the solution.
7. Find the solution of
for which 
Calculate the amplitude (a),
and the phase (d) of the solution.
11.3 The Non-homogeneous Equation. Forced Oscillations
11.3.1 General Remarks: Before proceeding to the solution of the problem when an external force f(t) is present, i.e., to the solution of the non-homogeneous equation, we make the observations:
If w and v are two solutions of the non-homogeneous equation, their difference u = w - v satisfies the homogeneous equation; we see this at once by substitution. Conversely, if u is a solution of the homogeneous equation and v a solution of the non-homogenous equation, then w = u + v is also a solution of the non-homogeneous equation. Hence we obtain from one solution - the particular integral - of the non-homogeneous equation all its solutions by adding the complete integral of the homogeneous equation - the complementary function. Hence we must only find a single solution of the non-homogeneous equation. Physically speaking, this means that, if we have a forced oscillation due to an external force and superpose on it an arbitrary free oscillation, represented by a solution of the homogeneous equation, we obtain a phenomenon which satisfies the same non-homogeneous equation as the original forced oscillation. If a frictional force is present, the free motion in the case of oscillatory motion fades out as time goes on, due to the damping factor e-rt/2m. Hence, for a given forced vibration with friction, it is immaterial what free vibration we superpose; the motion will always tend to the same final state as time goes on.
Secondly, we note that the effect of a force f(t) can be split up in the same way as the force itself. We mean by this: If f1(t) and f2(t) and f(t) are three functions such that
and if x1 = x1(t)
is a solution of the differential equation 
and x2 =
x2(t) is a solution of the equation 
, then x(t)=x1(t)+x2(t)
is a solution of the differential equation 
. Naturally, a
corresponding statement holds if f(t) consists
of any number of terms. This simple but important fact is called
the principle of superposition. The proof follows from a
glance at the equation itself. By subdividing the function f(t)
into two or more terms, we can thus split the differential
equation into several equations, which in certain circumstances
may be easier to manipulate.
The most important case is that of a periodic external force f(t). Such a periodic external force can be resolved into purely periodic components by expansion in a Fourier series, and, provided it is continuous and sectionally smooth - the only case of importance in physics -, it can be approximated as closely as we please by a sum of a finite number of purely periodic functions. Hence it is sufficient to find the solution of the differential equation subject to the assumption that the right hand side has the form
where a, b and w are arbitrary constants.
Instead of working with these trigonometric functions, we can obtain the solution more simply and neatly by using complex notation. We set f(t) = ceiwt and the principle of superposition shows that we need only consider the differential equation
where c is an arbitrary real or
complex constant. Such a differential equation represents
actually two real differential equations. In fact, if we split
the right hand side into two terms, e.g., if we take c =
1 and write 
then x1 and x2
- the solutions of the two real differential equations
combine to form the solution x = x1+ ix2 of the complex differential equation. Conversely, if we first solve the differential equation in complex form, the real part of the solution yields the function x1 and the imaginary part the function x2.
11.3.2 Solution of the
Non-homogeneous Equation: We will solve
the equation by a naturally suggested device. We assume that c
is real and (for the time being) that r ¹ 0. We now guess
that there will exist a motion which has the same rhythm as the
periodic external force, and attempt accordingly to find a
solution of the differential equation in the form
where we need only determine the factor s, which is
independent of the time. If we substitute this expression and its
derivatives 
into the
differential equation and remove the common factor eiwt,
we obtain the equation
Conversely, we see that for this value of s, the
expression 
is actually
a solution of the differential equation. However, in order to
express the meaning of this result clearly, we must perform a few
transformations.
We start by writing the complex factor s in the form
where the positive distortion factor a and the phase displacement are expressed in terms of the given quantities m, r k by the equations
In this notation, the solution assumes the form
and the meaning of the result is: There corresponds to the force c coswt, the effect cacosw(t - d) and to the force c sin wt the effect casinw(t - d).
Hence we see that the effect is a function of the same type as the force, i.e., an undamped oscillation. This oscillation differs from the oscillation representing the force in that the amplitude is increased in the ratio a : 1 and the phase is altered by the angle wd. Of course, it is easy to obtain the same result without using complex notation, but at the cost of somewhat longer calculations.
According to the remark at the beginning of this section, we have solved completely the problem by finding this one solution; in fact, by superposition of any free oscillation, we can obtain the most general forced oscillation.
Collecting our results, we can make the statement:
The complete integral of the differential equation
(where x ¹ 0) is x = caeiw(t-d) + u, where u
is the complete integral of the homogeneous equation 
and the
quantities a and d are
defined by the equations
The constants in this general solution allow us
to make the solution suit an arbitrary initial state, i.e., for
arbitrarily assigned values of 
the constants can be
chosen in such a way that 
11.3.3 The Resonance Curve: In order to acquire a grasp of the above solution and of its significance in applications, we shall study the distortion factor a as a function of the exciting frequency w, i.e., the function
The reason for this detailed study is that for
given constants k, m, r or, as we say,
for a given oscillatory system, we can think of the system as being acted upon by
periodic exciting forces of very different circular frequencies,
and it is important to consider the solution of the differential
equation for these widely different exciting forces. In order to
This
number w0 is the circular frequency which the system
would have for free oscillations, if the friction r were
zero or, more briefly, the natural frequency of the
undamped system. Owing to the
friction r, the actual frequency of the free system is
not equal to w0, but is instead
where we assume that 4km - r² > 0. (If this is not the case, the free system has no frequency; it is aperiodic.)
The function f(w) tends
asymptotically to the value 0 as the exciting frequency tends to
infinity, and, in fact, it vanishes to the order 1/w².
Moreover, f(0)=1/k; in other words, an exciting force of
frequency zero and magnitude 1, i.e., a constant force of
magnitude 1 causes the displacement of the oscillatory system to
rise to 1/k. In the region of positive values of w, the
derivative f '(w) cannot vanish except when the derivative of the
expression 
vanishes, i.e., for a value w = w1
>0, for which the equation
holds. In order that such a value may exist, we must obviously have in this case 2km - r²>0 ,
Since the function f(w) is positive everywhere, increases monotonically for small values of w and vanishes at infinity, this value must yield a maximum. We call the circular frequency w1 the resonance frequency of the system.
By substituting this expression for w1, we find that the value of the maximum is
As r ® 0, this value increases beyond all bounds. For r = 0, i.e., for an undamped oscillatory system, the function f(w) has an infinite discontinuity at the value w = w1. This is a limiting case to which we shall give special consideration below.
The graph of the function f(w) is called the resonance curve of the system. The fact that, forw=w1 (and consequently for small values of r in the neighbourhood of the natural frequency), the distortion of the amplitude a=f(w) is particularly large is the mathematical expression of the phenomenon of resonance, which for fixed values of m and k is more and more evident as r becomes smaller and smaller.
We have sketched in Fig. 3 below a family of
resonance curves, all of them corresponding to the values m
= 1 and k = 1, and consequently to w0 = 1,
but with different values of D = ½r. We see
that, for small values of D, there occurs well marked
resonance near w = 1; in the limiting case D=0, there would be
an infinite discontinuity of f(w) at w = 1, instead of a maximum. As D increases, the
maxima move towards the left andm, for the value 
, we have w1=0.
In this last case, the point, where the tangent is horizontal,
has moved to the origin and the maximum has disappeared. If 
,
there is no zero of f '(w); the resonance curve has no longer a maximum and there
does not occur any longer any resonance.
In general, the resonance phenomenon ceases as soon as the condition
becomes true. In the case of the equality sign, the resonance curve reaches its largest height f(0)=1/k at w1 = 0; its tangent is horizontal there and after an initial course, which is almost
horizontal, it drops towards zero.
11.3.4 Further Discussion of the
Oscillation:. However, we
cannot be content with the above discussion. In order that we may
really understand the phenomenon of forced motion, an additional
point must be emphasized. The particular integral 
is to be regarded as a limiting state
which the complete integral
approaches more and more closely as the time advances,
since the free oscillation 
superposed on the particular integral
fades away with passing time. This fading away will take place
slowly, if r is small, rapidly if r is large.
For example, let at the beginning of the motion, i.e., at time t = 0, the system be at rest, so that
We can determine from this condition the
constants c1 and c2, and
we see at once that not both are zero. Even when the exciting frequency is
approximately or exactly equal to w1, so
that resonance occurs, the relatively large amplitude a
= f(w1) will at first not appear. On the contrary,
it will be masked by the function 
and will first make its
appearance when this function fades away, i.e., it will appear
more slowly with a smaller r.
For the undamped system, i.e., for r = 0, our solution fails when the
exciting frequency is equal to the natural circular frequency 
because
then f(w0) is infinite. We therefore cannot obtain a
solution of the equation 
in the form s eiw t. However, we can at once obtain a
solution of the equation in the form s teiw t. If we substitute this expression
into the differential equation, remembering that
we have
and, since mw² = k,
Thus, when resonance occurs in an undamped system, we have the solution
In real notation, when f(t) =
coswt, we have 
, and when f(t)=sin
wt, we find
Thus, we see that we have found a function which may be referred to as an oscillation, but the amplitude of which increases proportionally with the time. The superposed free oscillation does not fade away, since it is not damped; but it retains its original amplitude and becomes unimportant in comparison with the increasing amplitude of the special forced oscillation. The fact that in this case the solution oscillates backwards and forwards between positive and negative bounds, which continually increase as time goes on, represents the real meaning of the infinite discontinuity of the resonance function in the case of an undamped system.
11.3.5. Remarks on the Construction of Recording Instruments:. For a great variety of applications in physics and engineering, the preceding discussion is of the utmost importance. With many instruments such as galvanometers, seismographs, oscillatory electrical circuits in radio receivers and microphone diaphragms, the problem is to record an oscillatory displacement, due to an external periodic force. In such cases, the quantity x satisfies our differential equation at least in first approximation.
If T is the period of oscillation of the external periodic force, we can expand the force in a Fourier series of the form
or, better still, we can think of it as
represented with sufficient accuracy by a trigonometric sum 
consisting of a finite number of terms. By the principle of superposition, the solution x(t) of the
differential equation, apart from the superposed free
oscillation, will be represented by an infinite series (the
convergence of which will not be discussed here) of the form
or approximately by a finite expression of the form
By virtue of our previous results,
We can then describe the action of an arbitrary periodic external force in the following ways: If we analyze the exciting force into purely periodic components - the individual terms of the Fourier series -, then each component is subject to its own distortion of amplitude and phase displacement, and the separate effects are then superposed by addition. If we are only interested in the distortion of amplitude (the phase displacement is only of secondary importance in applications and, moreover, can be discussed in the same way as the distortion of amplitude, for example, it is imperceptible to the human ear), a study of the resonance curve yields complete information about the way in which the motions of the recording apparatus reproduce the external exciting force. For very large values of l or w ( = 2p lt/T), the effect of the exciting frequency on the displacements x will be hardly perceptible. On the other hand, all exciting frequencies in the neighbourhood of w1, the (circular) resonance frequency, will markedly affect the quantity x.
In the construction of physical measuring and recording apparatus, the constants m, r and k are at our disposal, at least within wide limits. These should be chosen so that the shape of the resonance curve is as well adapted as possible to the special requirements of the measurement in question. Here two considerations predominate. In the first place, it is desirable that the apparatus should be as sensitive as possible, i.e., for all frequencies w in question, the value of a should be as large as possible. For small values of w, as we have seen, a is approximately proportional to l/k, so that the number 1/k is a measure of the sensitivity of the instrument for small exciting frequencies. The sensitivity can therefore be increased by increasing l/t, i.e., by weakening the restoring force.
The other important point is the necessity for
relative freedom from distortion. Let us assume that the
representation 
is an adequate approximation to
the exciting force. We then say that the apparatus records the
exciting force f(t) with relative freedom
from distortion, if, for all circular frequencies w £ N 2p/T,
the distortion factor has approximately the same value. This
condition is indispensable, if we wish to derive conclusions
about the exciting process directly from the behaviour of the
apparatus, for example, if a gramophone or wireless set is to
reproduce both high and low musical notes with an approximately
correct ratio of intensity. The requirement that the reproduction
should be relatively distortionless can never be satisfied exactly, since no portion of the
resonance curve is exactly horizontal. However, we can attempt to
choose the constants m, k and r of the
apparatus in such a way that no marked resonance occurs, and also
in such a way that the curve has a horizontal tangent at the
beginning, so that j(w) = a remains approximately constant for small values of w. As we
have learnt above, we can do this by setting
Given a constant m and a constant k, we can satisfy this requirement by adjusting the friction r properly, e.g., by inserting a properly chosen resistance in the electrical circuit. The resonance curve then shows that, from the frequency 0 to circular frequencies near the natural circular frequency w0 of the undamped system, the instrument is nearly distortionless and that above this frequency the damping is considerable. Hence we obtain relative freedom from distortion in a given frequency interval by first choosing m so small and k so large that the natural circular frequency w0 of the undamped system is larger than any of the exciting circular frequencies and then choosing a damping factor in accordance with the equation 2km - r².
For the following equations find the solution,
satisfying the initial conditions 
For Equations 1- 4 state also the amplitude, the
phase and the value of w for which the amplitude is a maximum:
11.4 Additional Remarks on Differential Equations
A more systematic study of differential equations is made Volume II, Chapter VI. We will present here only a few additions to the preceding special theory.
11.4.1 Homogeneous Linear Differential Equations of Order n with Constant Coefficients: More complicated vibration problems lead to linear differential equations for the unknown functions x(t) of the independent variable of the form
where a1, ··· , an are constants and n is a positive integer. We can solve this equation by a method similar to that for the case n = 3 (cf. 11.2). Let x = elt. If we substitute this function and its derivatives into the differential equation and remove the common factor elt, we obtain the equation of the n-th degree in l:
If l is a root of this equation, elt satisfies the differential equation. We shall now examine the various possibilities. Let l1, l2, ··· , ln be the roots of the equation f(l) = 0, so that
First of all, let all the roots differ. If all
the ln are real, then we obtain n
linearly independent solutions 
just as before.
The general solution is any linear combination
of these solutions. The constants cn can be determined so that x and its first n-1 derivatives take arbitrarily pre-assigned values at time t = 0; in order to do this, we must solve the system of n linear equations:
This set of equations has always a solution, if the roots are not equal, because the determinant of the coefficients is not zero.
If two of the roots are equal, say l1
= l2, then not only 
is a solution. This can be verified as
follows: since f(l) = 0 has a double root l=l1=l2,
if follows by a well-known theorem of algebra that
Now, by Leibnitz's rule for the derivative of a product,
Substituting in the differential equation, we find
since f(l) = 0 and, by the above remark regarding double roots, f '(l) = 0.
In the same manner, if l1, l2, ··· , ln are equal, we obtain the linearly independent solutions:
which may be combined to give a general solution depending on c1, c2, ··· , cn. These parameters again enable us to adapt the solution to n pre-assigned conditions, so that, for t = 0, we can fix the value of x(0) and its first n - 1 derivatives.
If the equation has complex roots, then, by a theorem of algebra, the roots occur in pairs, each one together with its conjugate root. Just as in the case n=2, we obtain solutions of the form
A few examples will serve to illustrate the above results.
The general solution is
A particular solution for which x = 2,
x' = 0 at t = 0 is 
The general solution is 
The general solution is 
11.4.2 Bernoulli Equation: An equation of the type
where A and B are only functions of t, is called a linear equation. In the case B = 0, if x=a(t), x=b(t) are solutions, any linear combination of a and b is also a solution. We shall now consider the slightly more general type of equation
where n is a positive integer; it is called Bernoulli equation. First, consider the simpler case where B is zero, i.e., where
Rewriting the equation in the form 
we see
that it can be integrated immediately:
if we write ec = v.
We now try to satisfy Bernolli' equation by a
function of the form 
where we assume that v is a
variable, so that
On substitution, we find
which can be integrated at once, so that
The above method is very important and may be applied in many cases. It is called variation of parameters; further details are given in Volume 2. Note that our solution is expressed in terms of integrals which, in general, cannot be expressed in terms of the elementary functions.
Example: Consider the equation
Let
then
and the eqnation becomes
By integration, we find
This result could have been obtained by direct substitution in the formula given above, but actually carrying out the method is far more instructive.
11.4.3 Other First Order Differential Equations Solvable by Simple Integration: There are a few other types of first order differential equations which can be solved by integration (although in most cases the integration cannot be performed explicitly in terms of elementary functions).
The first method we consider is that of separation of variables. If the differential equation can be given the form
i.e., y'B(y) + A(x) = 0, the variables are said to be separable. Obviously, the solution is
Example: Consider the equation
Here,
whence
Another type of equation, which can be solved, is of the form
where M and N are homogeneous functions of x and y of the same degree. In this case, the fraction M/N is a function of y/x only and we may write
If we set y = xv, this becomes
The variables x, v are now separable as shown:
Integrating, we have
Example: Consider the equation
Substituting y = vx, we have
Now, integration yields
11.4.4 Differential Equations of the Second Order: There are a few types of non-linear differential equations the solutions of which can also be found by integration. One type has already been discussed implicitly in 5.4.4 in our study of the motion of a particle on a given curve. This type is as follows:
We set v = dx/dt, so that
and our equation becomes
This equation may be regarded as one of the first order with v as dependent and x as independent variable. Separating the variables and integrating, we have
Then
which can be solved by integration (although, in general, it is impossible to carry it out explicitly).
This device lets us solve equations of the types:
which, if we set v = dx/dt, reduce to
These are equations of the first order which may be solvable by the preceding methods. This solution, after v has been replaced by dx/dt, will again be a differential equation of the first order, which must be solved for x. A few examples will make the process clear.
Examples:
1. 
Setting dy/dx = p, the equation becomes
Integration by separation of variables yields
integration
squaring
2.
Setting dy/dx = p yields
integration
whence
3. 
Set dy/dx = p, then dy/dx = pdp/dx. Now
Integration yields
whence
Integration yields
Exercises 11.3: Solve the differential equations
23. Find the motion of a particle moving along a straight line under the attraction of a force varying with the inverse square of the distance from the origin.
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