60.3.222 problem 1226

Internal problem ID [11232]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1226
Date solved : Tuesday, January 28, 2025 at 05:42:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-v \left (v -1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.124 (sec). Leaf size: 23

dsolve((x^2+1)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-v*(v-1)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 30

DSolve[(1 - v)*v*y[x] + 2*x*D[y[x],x] + (1 + x^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {LegendreP}(v-1,i x)+c_2 \operatorname {LegendreQ}(v-1,i x) \]