# 2 ODE questions

• Dec 8th 2010, 03:55 AM
MuhTheKuh
2 ODE questions
How do I show, that (x, 1+x²) is a fundamental set for the equation
(1-x²)*(d²y/dx²)+2x*(dy/dx)-2y=0
and state an open interval where I belongs to R on which y=cx+d(1+x²) is the unique solution to the equation (c and d being arbitrary constants)

and

how do I find the general solution to
(d^4y/dx^4)+8(d²y/dx²)-9y=e^x-sin(x)

• Dec 8th 2010, 04:57 AM
Mmmm
For your second problem, just treat it as you would a 2nd order ODE
ie solve the homogeonous eqn y''''+8y''-9y=0 by trying y=me^x.
This gives you a quartic auxilliary eqn to solve which in turn gives you the complementary function.

then find the particular integral by trying the solution y=Exe^x+Fsinx+Gcosx to find E,F,G.

add the complementary function to the particular integral to get the general solution.

Can't help with your first problem, sorry...
• Dec 8th 2010, 06:22 AM
Ackbeet
For your first problem, just show that x and 1+x² are both solutions to the DE, and then use the Wronskian to show linear independence.