Initial value problem Euler equation

[lisa92]

Question:

Find y as a function of x:

x^2 y'' + 8 x y' - 18 y = x^8

y(1)=3, y'(1)=2


Attempted solution:

I found the general equation to be Ax^(-9)+Bx^2+Cx^8.
However when I try to solve the initial value problem for this equation I have 3 unknowns.

 

[I like Serena]

Welcome to PF, lisa92!

Did you substitute your solution in the DE?
If you do, you'll find you have a 3rd equation.

 

[lurflurf]

write
(x²D²+8 x D-18)=(xD+9)(xD-2)
or change variables u=x² y

your general solution is extraneous check it in the equation to eliminate

 

[HallsofIvy]

That's an "Euler type" equationl. The change of variable x= ln(t) changes it to a differential equation with constant coefficients which may be simpler.

If you do not wish to do it that way, what did you get as a solution to the associated homogeneous equation?

 

[SteamKing]

Check out Euler-Cauchy equations at the bottom of this link:

http://sosmath.com/tables/diffeq/diffeq.html

This will help you obtain the solution to the homogeneous equation.

The particular solution will probably be y = k * x^8 + y (homogeneous)