Bounday-Value Problem: Eigenvalue and Eigenfunctions

physicsfan24
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Homework Statement


This is the original question:
\frac{d^{2}y}{dx^{2}}-\frac{6x}{3x^{2}+1}\frac{dy}{dx}+\lambda(3x^{2}+1)^{2}y=0

(Hint: Let t=x^{3}+x)
y(0)=0
y(\pi)=02. The attempt at a solution
This might be all wrong, but this is all I can think of
\frac{dt}{dx}=3x^{2}+1

so \frac{d^{2}y}{dx^{2}}-\frac{6x}{\frac{dt}{dx}}\frac{dy}{dx}+\lambda(\frac{dt}{dx})^{2}y=0After this, I do not know how to proceed to eliminate d^{2}y/dx^{2}, much less what else to do. Help!
Thank you very much for your time,
-PhysicsFan24
 
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holy **** bro are you in my class, MIAMI DADE DEs?? LOL and were both on here lookin 4 help here.

check out my thread, its the whole paper lol. yo you got the answers for any of the others??
 
Umm, I'm in U of Virgina and this is an online HW question... Yes I am taking ODE. You're in my class?
 
nevermind. I am in Miami Florida. But I got the same question as you at the same time. One hell of a coincidence. Let me know if you find the answer, and you can check my thread too, I got the same question posted on there.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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