Solving the Euler Cauchy Equation: Finding the General Solution

[engineer_dave]

Homework Statement

Find the general solution of x^2y" - 2y = 0

Homework Equations

The Attempt at a Solution

Can anyone tell me how to find the general solution of the Euler Cauchy equation. How do we make it into one?? Thanks.

 

[quasar987]

there is a general method for solving 2nd order ODE of the form y''+f(x)y+g(x)=0 that you can look up in any self respecting book on differential equations. for instance boyce & diprima

 

[HallsofIvy]

Homework Statement

Find the general solution of x^2y" - 2y = 0

Homework Equations

The Attempt at a Solution

Can anyone tell me how to find the general solution of the Euler Cauchy equation. How do we make it into one?? Thanks.

Why would you be given the problem of solving an Euler-Cauchy equation if you were told nothing beforehand about solving such a thing?

Try a "trial solution" of the form y= x^r where r is an unknown number.

 

[Kummer]

The characheristic equation here is $$k(k-1)-2=0$$. The solution would then be $$y=c_1|x|^{k_1}+c_2|x|^{k_2}$$ on $$(-\infty,0)\cup (0,\infty)$$.