Sturm-liuville prblm?

Question

Hi, I have math exam tomorrow and I have a SL exercise I was not able to solve ( (1-x^2)*y''(x)-x*y'(x)+(n^2)*y(x)=0 ), could someone be kind enough so to send me an explanation about how to solve this kind of problems and the solution (with the explanation) of the problem to my e-mail: [e-mail], I give you the points but please help, I'm so lost in that class that I realy need to have a good grade in the SL problems

Thanks

the equation is:( ...........(1-x^2)*y''(x)-x*y'(x)+(n^2)*y(x)=0......... ),

(1 - x ^ 2 ) * y''(x) -x * y'(x) + (n^2) * y(x) = 0

Answer 1

Let y’=yz, then y’’ = y’z+yz’ = yzz+yz’ and (1-xx)*(yzz+yz’) –xyz +nny =0,
hence y{(1-xx)(zz+z’) – xz +nn} =0;
y=0 is a trivial solution and is not interesting.

So (1-xx)(zz+z’) – xz +nn = 0; z’ -xz/(1-xx) +nn/(1-xx) =0 (eq1);
Consider first: z’ –xz/(1-xx) =0, then z’/z = x/(1-xx), hence ln|z| = u -0.5ln|1-xx| or z= u/sqrt(1-xx), where u is integration constant;
Now let u be also function of x, then z’ = u’/sqrt(1-xx) +ux/(sqrt(1-xx))^3; input z and z’ to (eq1),
then u’/sqrt(1-xx) +ux/(sqrt(1-xx))^3 – xu//(sqrt(1-xx))^3 +nn/(1-xx) = 0;
thence u’ = -nn/sqrt(1-xx), hence u = nn*acos(x)+p, p is integration constant;
thence z = y’/y = (nn*acos(x) + p) / sqrt(1-xx);
hence ln|y/q| = 0.5(n*arcos(x))^2 –p*acos(x), where q is integration constant;
thus y = q*exp(0.5(n*arcos(x))^2 –p*acos(x));
I’m sure a mistake or two have sneaked into solution, so check me by differentiation, general approach being helpful though.

Answer 2

Put spaces in the equation then we can read it!

With no spaces 219947907957939509855988958958955897597999053982829589208955893589398530

but with spaces 2199479 0795793 950985 598895 8958955 89759799 90539 82829589 208955 893589 398530