60.3.307 problem 1313

Internal problem ID [11317]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1313
Date solved : Tuesday, January 28, 2025 at 05:58:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.293 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.255 (sec). Leaf size: 75

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

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