60.3.374 problem 1380

Internal problem ID [11384]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1380
Date solved : Monday, January 27, 2025 at 11:18:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 67

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

\[ y = \sqrt {x \left (a -x \right )}\, \left (\left (\frac {x}{a -x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{2} +\left (\frac {a -x}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.239 (sec). Leaf size: 112

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

\[ y(x)\to \exp \left (\int _1^x\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[2]+c_1\right ) \]