This one has me running in circles. Find the General Solution to y''+xy'-(1+x)y=0 !!!
I have heard that using variation of parameters helps but isn't that for a problem = to some f(x)? I have tried substituting y=e^u and canceling e^u which gets you 1+u-(1+u)=0 does this mean its a double root of 1? thank you for any insight, greatly appreciated.

Question

This one has me running in circles.  Find the General Solution to y''+xy'-(1+x)y=0 !!!
 I have heard that using variation of parameters helps but isn't that for a problem = to some f(x)?  I have tried substituting y=e^u and canceling e^u which gets you 1+u-(1+u)=0   does this mean its a double root of 1? thank you for any insight, greatly appreciated.

anonymous · Answer

can you use the integrating factor method? It works for a function in the form $$\frac{dy}{dx} + g(x)y = h(x)$$ not sure about second derivatives.

anonymous · Answer

not sure if it works for second derivatives..
I was on the wrong track before, but using reduction of order method works out. Thanks for replying!

anonymous · Answer

np sorry i couldnt help

anonymous · Answer

Laplace transforms would also work :)

anonymous · Answer

Assuming you have initial conditions