
2 ODE questions
How do I show, that (x, 1+x²) is a fundamental set for the equation
(1x²)*(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^xsin(x)
Thanks for your time

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...

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.