- #1
Illusionist
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Homework Statement
Given that y=x^2 is a solution to the differential equation:
(x^2)y'' + 2xy' - 6y = 0 <--- Eq.(1)
find the general solution of the differential equation
(x^2)y'' + 2xy' - 6y = 10(x^7) + 15(x^2) <--- Eq.(2)
Hence write down a second linear dependent solution of equation (1) and a particular solution of equation (2).
Homework Equations
I've basically concluded that variation of parameters is necessary. I don't think I completely understand what is being asked.
The Attempt at a Solution
I tried letting y= V(y1) = V(X^2)
hence y'= (x^2)V' + 2xV and
y''= (x^2)V'' + 2xV' + 2V
Here is where I think I'm getting confused, sub. back into (1) I get:
V''(x^4)+2(x^3)V'+2V(x^2)+2(x^3)V'+4(x^2)V-6V(x^2)=0
which equals V''(x^4)+4(x^3)V"=0
This is where I come to a dead end, any help or advice would be greatly appreciated, thank you.