# DE I don't get

1. Use substitution x = e^t to transfer equation into one with constant coefficients and solve:
x^2 y'' -3xy' + 13y = 4 + 3x
My work:
Okay, we have
e^2t y'' - 3e^t y' + 13y = 4 + 3e^t.

Now I am absolutely stuck. No idea how to solve it. Help!

HallsofIvy
Homework Helper
You didn't complete the substitution! Your y' and y" are still with respect to x, not t.

Use the chain rule: dy/dx= dy/dt dt/dx. Since x= et, t= ln x and
dt/dx= 1/x. That is, dy/dx= (1/x)(dy/dt).
d2y/dx2= d((1/x)(dy/dt)/dx= (-1/x2)dy/dt+ (1/x)d(dy/dt)/dx= ((-1/x2)dy/dt+(1/x)(1/x)d2y/dt2=.
Now, x2 y'' = d2y/dt2- dy/dt
and -3xy'= -3 dy/dt
so the differential equation is
d2y/dt2- 4dy/dt+ 13y= 4+ 3ett

That should be easy to solve!