Solve the given differential equation by undetermined coeffi

Solve the given differential equation by undetermined coefficients. 1/4y\" = y\' + y = x^2 - 2x y(x) =

Solution

First we solve assocaited homogeneous ode

y\'\'/4+y\'+y=0

y\'\'+4y\'+4y=0

It is a linear homogeneous ode with constant coefficients

SO y=exp(kx)

Substituting gives

k^2+4k+4=0

k=-2, repeated roots .

So general solution to homogeneous ode is

yh=e^{-2x}(Ax+B)

For particular solutoin based on inhomogeneous part

x^2-2x we make the guess

yp=Cx^2+Dx+E

yp\'=2Cx+D

yp\'\'=2C

Substitutint gives

C/2+2Cx+D+Cx^2+Dx+E=x^2-2x

COmparing coefficients gives

C=1

2C+D=-2

D=-5

C/2+D+E=0

1/2-5+E=0

E=9/2

yp=x^2-5+9/2

General solution is

y=yh+yp=e^{-2x}(Ax+B)+x^2-5+9/2