54.3.363 problem 1380

Internal problem ID [12658]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1380
Date solved : Wednesday, October 01, 2025 at 02:19:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 67

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

\[ y = \sqrt {x \left (-x +a \right )}\, \left (\left (\frac {x}{-x +a}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_2 +\left (\frac {-x +a}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_1 \right ) \]

✓ Mathematica. Time used: 0.148 (sec). Leaf size: 112

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

\begin{align*} 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 ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(b*y(x)/(x**2*(-a + x)**2) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE b*y(x)/(x**2*(-a + x)**2) + Derivative(y(x), (x, 2)) cannot be solved by the hypergeometric method