Solve appropriate linear differential equations using techni

Solve appropriate linear differential equations using techniques such as reduction of order, variation of parameters and series expansions. Question Obtain the general solution of y\" - 2/x^2y = 1/x, x > 0

Solution

We can rewrite the ode as

x^2y\'\'-2y=x

First we solve the associated homogeneous ode ie

x^2y\'\'-2y=0

LEt, y=x^m

Substituting gives

m(m-1)x^m-2x^m=0

m(m-1)-2=0

m^2-m-2=0

m^2-2m+m-2=0

m(m-2)+(m-2)=0

m=-1,2

HEnce, 1/x and x^2 are solutions to this homogeneous ode

So general solution to homogeneous ode is

yh(x)=A/x+Bx^2

Now we look for particular solution

Base on inhomogeneous part ie x

We make a guess for particular solution

yp=Ax

Substituting gives

-2Ax=x

A=-1/2

yp=-x/2

So general solution is

y(x)=yh+yp=A/x+Bx^2-x/2