60.3.364 problem 1370

Internal problem ID [11374]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1370
Date solved : Monday, January 27, 2025 at 11:18:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 19

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

\[ y = c_{1} \sinh \left (a \,\operatorname {arctanh}\left (x \right )\right )+c_{2} \cosh \left (a \,\operatorname {arctanh}\left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 2.982 (sec). Leaf size: 87

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

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {K[1]+\sqrt {a^2}}{K[1]^2-1}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+\sqrt {a^2}}{K[1]^2-1}dK[1]\right )dK[2]+c_1\right )}{\sqrt {x^2-1}} \]