60.3.350 problem 1356

Internal problem ID [11360]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1356
Date solved : Tuesday, January 28, 2025 at 06:04:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.286 (sec). Leaf size: 29

dsolve(diff(diff(y(x),x),x) = -(2*x^2+1)/x/(x^2+1)*diff(y(x),x)-(-v*(v+1)*x^2-n^2)/x^2/(x^2+1)*y(x),y(x), singsol=all)
 

\[ y = \operatorname {LegendreQ}\left (v , n , \sqrt {x^{2}+1}\right ) c_{2} +\operatorname {LegendreP}\left (v , n , \sqrt {x^{2}+1}\right ) c_{1} \]

✓ Solution by Mathematica

Time used: 0.340 (sec). Leaf size: 78

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

\[ y(x)\to c_1 x^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-v),\frac {1}{2} (-n+v+1),1-n,-x^2\right )+c_2 x^n \operatorname {Hypergeometric2F1}\left (\frac {n-v}{2},\frac {1}{2} (n+v+1),n+1,-x^2\right ) \]