14.1.1 Direct integration

If F(x, y) = f(x) is a function of x only, then (14.4) takes the form

(14.5a)

and may be directly integrated to give

where c is an arbitrary constant. If F(x, y) = f(y) is a function of y only, then we have

(14.5b)

so that the solution

is again obtained by direct integration.

14.1.2 Separation of variables

One frequently meets applications in which the right-hand side of (14.4) is a product F(x, y) = f(x)g(y), so that it becomes

(14.6a)

If either f or g is a constant, then (14.6a) can be evaluated by direct integration, as in Section 14.1.1. Otherwise, rearranging (14.6a) gives

that is, the variables have been separated. Integrating then gives

(14.6b)

which expresses y implicitly in terms of x. In doing the integration one must of course remember to include the constant of integration.

14.1.3 Homogeneous equations

In Section 7.2.4, a function f(x₁, x₂, …, x_n) was defined to be homogeneous of degree k if

where λ is an arbitrary parameter. Thus, for example, the function f(x, y) = x³y + x²y² is a homogeneous function of degree 4, whereas f(x, y) = xy + x/y is inhomogeneous because it does not satisfy this requirement. Homogeneous equations are of the form

(14.7)

where g and h are both homogeneous functions of the same degree. The key property is that the right-hand side of (14.7) can be written as a function of the ratio z ≡ y/x, i.e.

and

Substituting these equations into (14.7) gives

(14.8a)

This is a separable equation, which can be integrated to give

(14.8b)

from which z and hence y can be found.

Some equations, although not obviously homogeneous at first sight, may be reduced to this form by suitable transformation of variables. One type that commonly occurs is given in Example 14.3(b).

14.1.4 Exact equations

If

then the first-order equation (14.4) may be written

(14.9)

If it is possible to find a function f(x, y) such that

(14.10)

then, by Equation (7.17a),

is an exact differential and (14.9) is called an exact equation. We then have

and comparing this with (14.9), we see that the latter has the implicit solution

(14.11)

where c is an integration constant. To see whether a function f(x, y) that satisfies (14.10) exists, we note that (14.10) implies

and hence

(14.12)

so that (14.12) is a necessary condition for a relation of the form (14.10) to exist. It can also be shown to be a sufficient condition. If it is satisfied, then integrating A with respect to x at fixed y, and B with respect to y at fixed x gives the results

(14.13a)

and

(14.13b)

where f₁(y) and f₂(x) are arbitrary functions, which may be identified up to a constant by comparing (14.13a) and (14.13b). Alternatively, (14.13a) may be differentiated with respect to y at fixed x and compared to ∂f/∂y = B to determine c₁(y). The solution is then given by (14.11), as we shall illustrate by an example.

14.1.5 First-order linear equations

First-order linear ODEs are equations of the form

(14.14)

where p(x) and q(x) are given functions of x, or constants. If the equation is exact, it may be solved by the method of Section 14.1.4. If it is not of this form, that is, it is inexact, it may in principle be solved by multiplying by a function I(x), to be determined below, called an integrating factor. We then obtain

(14.15a)

and I(x) is chosen so that the left-hand side of (14.15a) is equal to d[I(x)y]/dx, i.e.,

(14.15b)

Equating the linear terms in y in this equation gives

which on integrating gives (providing y ≠ 0) the result

(14.16)

for the integrating factor. Finally, from (14.15a) and (14.15b),

and hence the general solution for y is given by

(14.17)

with I(x) given by (14.16).

14.2.1 Complementary functions

As defined above, complementary functions are the general solutions of reduced equations of the form

(14.22)

Equations like (14.22) are often called homogeneous.² These equations have the important property that if y₁(x) and y₂(x) are solutions, then any linear combination

(14.23)

is also a solution. This result follows directly on substituting (14.23) into the left-hand side of (14.22). In what follows, we shall discuss second-order equations in some detail, because they are by far the most important in physics applications; then we briefly address the extension to higher-orders.

The second-order homogeneous equation is

(14.24)

where we have relabelled the constants for later convenience. As a trial solution we will take

(14.25)

since with this form, differentiating y just multiplies it by a constant m. Substituting (14.25) in (14.24) gives

and (14.25) is a solution of (14.24) when the bracket vanishes, that is, when m is a root of the auxiliary equation

(14.26)

This has roots

(14.27)

and three cases must be distinguished.

1. If b² > 4ac, there are two real roots m₁ and m₂, with m₁ ≠ m₂. The general solution of (14.24) is then, by the superposition principle (14.23),

   (14.28a)

   where A₁ and A₂ are arbitrary constants.

2. If b² = 4ac, then

   (14.29)

   so that (14.28a) would contain only one arbitrary constant A ≡ A₁ + A₂ and so cannot be the general solution. In this case, a second solution is obtained by writing

   

   where u(x) is a function to be determined. Substituting into (14.24) then gives

   (14.30)

   However, using (14.29), we see that

   

   so that the second term in (14.30) vanishes and we are left with the equation d²u/dx² = 0, with solution

   

   where again A₁ and A₂ are arbitrary constants. The general solution of (14.24) when the auxiliary equation has two equal roots is therefore

   (14.28b)

3. Finally, if b² < 4ac, there are two complex solutions

   (14.31)

   where α = −b/2a and . Nonetheless, a real solution of (14.24) is obtained by writing

   

   where A is a complex constant. Using (14.31) and e^z = cos z + isin z for any z, this can be rewritten in the form

   (14.28c)

   where and are two arbitrary real constants.

   This exhausts the types of solution for homogeneous second-order linear equations with constant coefficients.

   In the cases of higher order equations (14.20a), the substitution y = e^mx leads to the auxiliary equation

   (14.32)

   This gives n roots,³ where degenerate and complex roots are treated in the same way as in the second-order case. In particular, if k roots m₁, m₂, ⋅⋅⋅, m_k = m coincide, then the corresponding term in the solution, analogous to (14.28b), becomes

   (14.33)

14.2.2 Particular integrals: method of undetermined coefficients

We now turn to the case where f(x) ≠ 0 in (14.19), again restricting the discussion to the case of constant coefficients. We have already stated that the general solution of such an equation is the sum of the complementary function and a particular integral. The standard method for finding the complementary function has already been discussed, so it only remains to find the particular integral. Unfortunately, there is no general method for doing this, but there is a variety of methods, each of which is appropriate for a range of functions f(x). We will discuss two, starting with that known as the method of undetermined coefficients.

This method consists of assuming a trial form for the particular integral Y(x) that resembles f(x), but contains a number of free parameters. The trial function is then substituted into the differential equation and the parameters determined so that the equation is satisfied. It is most useful when f(x) contains only polynomials, exponentials, or sines and cosines. The rules for constructing the appropriate trial functions are as follows.

1. If f(x) = ae^bx where a and b are constants, the trial function is Y(x) = Ae^bx, where A is a constant.
2. If f(x) = asin px + bcos px, where a, b and p are constants (a or b may be zero), the trial function is Y(x) = Asin px + Bcos px, where A and B are constants.
3. If , where some of the coefficients may be zero, the trial function is , where the b_n are constants.
4. If any term in these trial functions is contained within the complementary function, then the trial function must be multiplied by the smallest integer power of x such that it then contains no term that is in the complementary function.
5. Finally, if f(x) is the sum or product of any of these forms, the trial function must be taken to be the sum or product of the appropriate individual trial functions.

*14.2.3 Particular integrals: the D-operator method

There are other methods for finding particular integrals, and by way of contrast we will discuss here the so-called D-operator method. This method has the advantage that it is not necessary to guess a trial function and the numerical coefficients multiplying the functional form of the particular integral are obtained automatically. However, it does require some experience in identifying which manipulations to use to obtain the solution.

The quantity D, defined by D ≡ d/dx, was introduced in Chapter 9, Section 9.3.2. It is a differential operator. That is, it only has a meaning when it acts on a function, which is always written to the right of D. Higher differential operators may be constructed from D. For example,

etc. We see that D satisfies the usual rules of algebra and so may be formally treated as an algebraic quantity, despite the fact that it cannot be evaluated as such to yield a numerical value. From the algebraic rules of differentiation it follows that if f(x) and g(x) are differentiable functions,

(14.36a)

and

(14.36b)

where c is a constant, and hence D is a linear operator (see Section 9.3.2).

In terms of D we may now write (14.19) as

(14.37a)

where

(14.37b)

is a polynomial operator in D of order n. Just as D may be formally treated as an algebraic quantity, so may a polynomial function of D, such as F(D), and in particular it may, in suitable cases, be factorised. Moreover the order of the factors is irrelevant provided the coefficients in (14.37) are constants. Thus, for example,

However, the order is relevant if the coefficients are functions of x. For example,

whereas

so

A number of useful results may be derived when the function f(x) = e^kx, where k is a constant. For example, using the fact that Dⁿe^kx = kⁿe^kx, it follows that

(14.38a)

Similarly, since D²sin kx = −k²sin kx, it follows that

(14.38b)

and

(14.38c)

If the exponential is multiplied by an arbitrary function V(x), then by considering the terms D[e^kxV(x)], D²[e^kxV(x)], etc. in succession, the result (14.38a) may be generalised in a straightforward way to

(14.38d)

Finally, in Chapter 3 we defined indefinite integration as the inverse operation of differentiation. In an analogous way we now defined the inverse operator D^{− 1} by

(14.39)

with

(14.40)

Also by analogy with the notation Dⁿ for successive differentiations, we will use D^{− n} ≡ 1/Dⁿ for the operation of n successive integrations.

We now return to the solution of equations of the type (14.19). If these are rewritten in the form (14.37a) and (14.37b), then a particular integral can be obtained by writing

(14.41)

provided we can interpret and evaluate the right-hand side, using the techniques introduced above. To illustrate this, consider the equation

(14.42)

In the D-operator formalism, this is written

giving the particular integral

We now have to decide how to handle the polynomial in the denominators, using the relations (14.38). If we use (14.38c) with k = 1, we have

Now we can again use (14.38c) to give

as the desired form for the particular integral.

A different technique is illustrated by considering the equation obtained by replacing cos x in (14.42) by xe^2x. The particular integral is then

and using (14.38d) with k = 2 gives

The denominator can be expressed as partial fractions, to give

(14.43)

The point of this decomposition is that each fraction can be expanded as a binomial series

where higher derivatives are not required because when acting on x they will be zero. Using these expansions, (14.43) becomes

which is the desired result.

*14.2.4 Laplace Transforms

The D-operator method converts a differential equation to an algebraic equation for D. Another method that does something very similar is based on the use of Laplace transforms. The Laplace transform F(p) of a function f(x) is defined by

(14.44)

where the parameter p may in principle be complex, but in the discussion that follows we will take it to be real.

The Laplace transforms of many simple functions may be found by direct evaluation of the defining integral (14.44). Others may then be found by using easily proved general properties of the Laplace transform. These include: linearity,

where a and b are constants; and the shift theorem

If L[f(x)] = F(p), then L[e^axf(x)] = F(p − a).

A related version of this, called the translation property, is

where H(x) is the unit step function

For example, the translation property follows by considering the Laplace transform of

which has the same form as f(x) but moved a distance a along the x axis. If we let z = x − a, then

Using a combination of these properties, several useful examples of Laplace transforms may be obtained and are shown in Table 14.1. As an example, we shall derive the result for e^{− ax}cos (bx). Firstly,

Then, using the shift theorem,

We can also define an inverse Laplace transform, denoted L^{− 1}, such that

It follows that LL^{− 1} = 1 and because L is linear, so is L^{− 1}. Inverse transforms for some simple functions follow from the results of Table 14.1. Thus, since L[1] = 1/p, it follows that L^{− 1}[1/p] = 1, but to find inverse transforms in general requires the techniques of complex variable theory and we will not discuss them here.

In order to use Laplace transforms to solve differential equations, we will also need the transforms of the differentials of a function y(x). For example, consider the Laplace transform of dy(x)/dx. From the definition (14.44), this is

Integrating by parts, this is

providing ye^{− px} → 0 as x → ∞. So,

(14.45a)

Likewise,

so that

where

This procedure may be repeated to obtain expressions for derivatives of any order. These can be used to convert a differential equation into an algebraic equation in F(p), and if the inverse transform L^{− 1} can be found, the solution of the equation for F(p) can be inverted, and a solution of the original equation ODE results. An advantage of the method is that it can easily incorporate boundary conditions on the function and its first derivative, as can be seen from (14.45). Its use is illustrated in Example 14.10.

Fourier transforms can similarly be used to convert a differential equation for y(x) into an algebraic equation for its Fourier transform. However, in contrast to Laplace transforms, their use is restricted to functions that tend to zero as |x| tends to infinity sufficiently rapidly for (13.45) to be satisfied. In these cases, it can be a useful technique, as illustrated in Problem 14.19, but we will not pursue this here.

It frequently happens that part of the expression for which the inverse transform L^{− 1} is to be found contains the product of two Laplace transforms. In this the case we can use the method of convolutions that was discussed in Chapter 13, Section 13.3.4 for Fourier transforms. Thus if g_i(p) is the Laplace transform of f_i(p), that is, if

(14.46)

where i = 1,2, then

(14.47a)

and hence

(14.47b)

where the convolution integral is

To prove (14.47a), we start from the definition (14.46) and form the product

(14.47c)

where u and are dummy variables. Now letting , and rewriting in terms of the variables t and u, changes the limits on the integrals, giving

This corresponds to summing the vertical strips on Figure 14.2a. But from the work of Chapter 11, we know that we can equally sum the horizontal strips, as shown in Figure 14.2b, that is, we can reverse the order of integrations. Therefore the double integral may be written

and so (14.47a) follows, which completes the proof.