ODE: find general solution

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1. The problem statement, all variables and given/known data

Find the general solution of the following ODE:
x[tex]^{2}[/tex]Y''(x) + xY'(x) - CY(x)=0

where C is a constant.

2. Relevant equations



3. The attempt at a solution

First I want to do this in the case where C=0; this gives:
x[tex]^{2}[/tex]Y''(x) + xR'(x)=0

How do i solve this ODE? Any hints? Thank you.
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution
 

HallsofIvy

Science Advisor
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Please show some work! Do you know anything about "Euler type" equations? (Also called "equipotential" equations.)
 
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I didn't show any working because i got stuck at that point :)
I know how to find the general solution of ODEs but not when there are functions of x infront of the Y'' and Y'.
I don't know about Euler type Equations.

Thank you
 

gabbagabbahey

Homework Helper
Gold Member
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6
Have you learned the method of power series solutions yet?
 
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no.
This question is in the section of partial differential equation.
I have the two equations:
xY''(x) + xY'(x) - CY(x)=0
and
ZZ''(t)+CZ(t)=0
I can do the second one since there's no function of t infron of the Z. But the first one...I havent come across something like that.
 

gabbagabbahey

Homework Helper
Gold Member
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6
You MUST have been taught at least one method of solving such ODE's....what methods have you learned?
 
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the lecturer gave us an example. He found the general seperable solution of Laplace's Equation: V[tex]_{xx}[/tex]+V[tex]_{yy}[/tex]=0

Where the PDE was converted into the ODEs:
X''(x)-[tex]\lambda[/tex]X(x)=0
Y''(y)+[tex]\lambda[/tex]Y(y)=0

This is simple to solve.

My original question was: find the general solution of:

V[tex]_{xx}[/tex]+(1/x)V[tex]_{x}[/tex]+(1/x^2)V[tex]_{tt}[/tex]
then i said: let V(x,t)=Y(x)Z(t)
so:
V[tex]_{xx}[/tex]=Z(t)Y''(x)
V[tex]_{tt}[/tex]=Y(x)Z''(t)
I substituted these into the original:
V[tex]_{xx}[/tex]+(1/x)V[tex]_{x}[/tex]+(1/x^2)V[tex]_{tt}[/tex]
and divided by Z(t)Y(x) and then multiplied by x^2; to get:
x^2Y''(x)+xY'(x)-CY(x)=0
Z''(t)+CZ(t)=0
 
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sorry, the powers are supposed to be subscripts.
 

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