Understanding Engineering Mathematics

always work, but is a good start, and we will come back to exceptional cases
later. This approach is called the method of undetermined coefficients. It in fact
works for equations of any order, provided the coefficients are constant, as we are
assuming here.

Table 15.1

f(x) Trial solution

Polynomial of degreen Polynomial of degreen:
f(x)=anxn+an− 1 xn−^1 +··· yp=Lxn+Mxn−^1 +···

Exponential: Similar exponential
f(x)=ekx yp=Lekx

Sinusoidal function: Linear combination of similar sinusoidals:
f(x)=sinωxor cosωx yp=Lcosωx+Msinωx

‘Damped’ sinusoidal: Linear combination of similar damped sinusoidals
f(x)=ekxcosωxorekxsinωx yp=Lekxcosωx+Mekxsinωx

The ‘forcing functions’ f(x)given in Table 15.1 cover most cases of physical
interest at the elementary level. They can in fact be used for much more complicated
functions,f(x), using linearity and the techniques of Fourier analysis (➤ 517) for
example, so they are in fact more general than you might think. We will give
an example of each of them before discussing the complications of exceptional
cases.

Problem 15.17

Find a particular integral for the equation

y′′Y 2 y′Yy=xY 2

Hence determine the general solution. Find the particular solution

satisfying the initial conditionsy. 0 /=0,y′. 0 /=0.

For reasons that become clear later, we will always find the CF first. In this case you can
treat it as an exercise to check that

yc=(Ax+B)e−x

Now for the PI we note thatf(x)is in this case a first degree polynomial – i.e. a linear
function. We therefore try a solution of the same form:

yp=Lx+M,soyp′=L,yp′′= 0

Substituting in the equation gives

0 + 2 L+Lx+M=Lx+( 2 L+M)≡x+ 2

Solving this identity (50

➤

)gives

L= 12 L+M= 2