If we consider the equation (1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+2y=0,\ -1<x<1 how can we find the

Oberlaudacu

Oberlaudacu

Answered question

2021-12-20

If we consider the equation
(1x2)d2ydx22xdydx+2y=0, 1<x<1
how can we find the explicit solution, what should be the method for solution?

Answer & Explanation

Jillian Edgerton

Jillian Edgerton

Beginner2021-12-21Added 34 answers

As mentioned in comment, this ODE is Legendre differential equation for l=1
ddx[(1x2)dy(x)dx]+l(l+1)y(x)=0
The standard Frobenius method will give us the Legendre polynomials.
By inspection, it is sort of obvious y(x)=x is one solution for this ODE. In general, if we already know one solution to a second order homogenous ODE, we dont
porschomcl

porschomcl

Beginner2021-12-22Added 28 answers

There is one obvious particular solution which is y=c1x. The second one is much less obvious to me but, using a CAS, I found as general solution
y=c1x+c2(xtanh1(x)1)
I hope and wish this will give you some ideas. As said by achille hui, beside the solution in terms of Legendre polynomials, you can obviously use Frobenius method which leads to the solution I wrote.
RizerMix

RizerMix

Expert2021-12-29Added 656 answers

Assume u=(1x2)dydx. Try to calculate dudx

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