60.3.392 problem 1398

Internal problem ID [11402]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1398
Date solved : Tuesday, January 28, 2025 at 06:04:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.457 (sec). Leaf size: 69

dsolve(diff(diff(y(x),x),x) = -1/(x^2-1)*(3*x^2-1)/x*diff(y(x),x)-(x^2-1-(2*v+1)^2)/(x^2-1)^2*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.210 (sec). Leaf size: 72

DSolve[D[y[x],{x,2}] == -(((-1 - (1 + 2*v)^2 + x^2)*y[x])/(-1 + x^2)^2) - ((-1 + 3*x^2)*D[y[x],x])/(x*(-1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
 

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