60.3.305 problem 1311

Internal problem ID [11315]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1311
Date solved : Tuesday, January 28, 2025 at 05:58:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.436 (sec). Leaf size: 52

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

\[ y = c_{1} \operatorname {hypergeom}\left (\left [-\frac {v}{2}, \frac {1}{2}+\frac {v}{2}\right ], \left [\frac {1}{2}\right ], x^{2}+1\right )+c_{2} \sqrt {x^{2}+1}\, \operatorname {hypergeom}\left (\left [1+\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {3}{2}\right ], x^{2}+1\right ) \]

✓ Solution by Mathematica

Time used: 0.510 (sec). Leaf size: 61

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

\[ y(x)\to c_2 G_{2,2}^{2,0}\left (-x^2| \begin {array}{c} \frac {1-v}{2},\frac {v+2}{2} \\ 0,0 \\ \end {array} \right )+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {v}{2},\frac {v+1}{2},1,-x^2\right ) \]