Recall that the hyperbolic cosine function is defined by cos

Recall that the hyperbolic cosine function is defined by cosh(kx) = e^kx + e^-kx/2, where k is a constant. Find a general solution of the equation y\" - 4y = cosh(2x).

Solution

First we solve the associated homogeneous ode

y\'\'=4y

General solution to this is

yh=A exp(2x)+B exp(2x)

Now based on the formula for cosh(x) we see cosh(x) is solution to homogeneous ode.

So we make guess for particular solution

yp= Cx exp(2x)+Dx exp(-2x)

Substituting gives

C=1/8, D=-1/8

HEnce,

yp= (x exp(2x)-x exp(-2x))/8

Genral solution is

y=yh+yp