# {x^2y''-20y=0, y(1)=1, y'(1)=gamma}

• Oct 22nd 2013, 08:23 AM
shane18
{x^2y''-20y=0, y(1)=1, y'(1)=gamma}
Find (gamma) so that the solution of the initial value problem
has finite limit as x → 0.

I'm not sure I fully understand the problem. I can solve the homogeneous equation using the indicial equation, letting y = x^r:

r^2 - r - 20 = 0
r= -4 , 5

y=c1 * x^-4 + c2 * x^5

for y(1)=1, 1 = c1 + c2 => c1 = 1- c2

for y'(1) = gamma, gamma = -4c1 + 5c2 => gamma = -4 + 4c2 + 5c2

c1 = (-gamma/9 + 5/9)

c2 = (gamma + 4)/9

thus y= (-gamma/9 + 5/9)x^-4 + ((gamma + 4)/9)x^5

but that doesn't seem to address the bit about the finite limit of the gamma function.... and my prof wants the answer in "gamma =" .......
• Oct 22nd 2013, 09:20 AM
topsquark
Re: {x^2y''-20y=0, y(1)=1, y'(1)=gamma}
Quote:

Originally Posted by shane18
Find (gamma) so that the solution of the initial value problem
has finite limit as x → 0.

I'm not sure I fully understand the problem. I can solve the homogeneous equation using the indicial equation, letting y = x^r:

r^2 - r - 20 = 0
r= -4 , 5

y=c1 * x^-4 + c2 * x^5

for y(1)=1, 1 = c1 + c2 => c1 = 1- c2

for y'(1) = gamma, gamma = -4c1 + 5c2 => gamma = -4 + 4c2 + 5c2

c1 = (-gamma/9 + 5/9)

c2 = (gamma + 4)/9

thus y= (-gamma/9 + 5/9)x^-4 + ((gamma + 4)/9)x^5

but that doesn't seem to address the bit about the finite limit of the gamma function.... and my prof wants the answer in "gamma =" .......

Check your solution for x = 0. What term do you need to get rid of?

-Dan
• Oct 22nd 2013, 09:29 AM
shane18
Re: {x^2y''-20y=0, y(1)=1, y'(1)=gamma}
for x= 0, the term with x^ -4 would be undefined.

Am I to disregard the r= -4 solution to the homogeneous equation, then?

Also, do I define gamma in terms of x and y?
• Oct 22nd 2013, 03:28 PM
topsquark
Re: {x^2y''-20y=0, y(1)=1, y'(1)=gamma}
Quote:

Originally Posted by shane18
for x= 0, the term with x^ -4 would be undefined.

Am I to disregard the r= -4 solution to the homogeneous equation, then?

Also, do I define gamma in terms of x and y?

If the x^{-4} term is a problem then you need to find a gamma such that the term isn't there any more. What value of gamma would do that for you? (Hint: What is the coefficient of the x^{-4} term? Can you set that to zero?)

-Dan