60.3.441 problem 1448

Internal problem ID [11451]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1448
Date solved : Monday, January 27, 2025 at 11:22:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 77

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

\[ y = \sqrt {a^{2}-x^{2}}\, \left (\left (\frac {a -x}{a +x}\right )^{\frac {\sqrt {a^{2}-b^{2}}}{2 a}} c_{1} +\left (\frac {a -x}{a +x}\right )^{-\frac {\sqrt {a^{2}-b^{2}}}{2 a}} c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.286 (sec). Leaf size: 106

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

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