60.3.365 problem 1371

Internal problem ID [11375]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1371
Date solved : Tuesday, January 28, 2025 at 06:04:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.124 (sec). Leaf size: 37

dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-(-a^2-lambda*(x^2-1))/(x^2-1)^2*y(x),y(x), singsol=all)
 

\[ y = c_{1} \operatorname {LegendreP}\left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right )+c_{2} \operatorname {LegendreQ}\left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right ) \]

✓ Solution by Mathematica

Time used: 0.050 (sec). Leaf size: 48

DSolve[D[y[x],{x,2}] == -(((-a^2 - \[Lambda]*(-1 + x^2))*y[x])/(-1 + x^2)^2) - (2*x*D[y[x],x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 P_{\frac {1}{2} \left (\sqrt {4 \lambda +1}-1\right )}^a(x)+c_2 Q_{\frac {1}{2} \left (\sqrt {4 \lambda +1}-1\right )}^a(x) \]