54.3.364 problem 1381

Internal problem ID [12659]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1381
Date solved : Wednesday, October 01, 2025 at 02:19:12 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.105 (sec). Leaf size: 219

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

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

✓ Mathematica. Time used: 0.125 (sec). Leaf size: 280

ode=D[y[x],{x,2}] == c - (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 (\int _1^x-c \exp \left (\int _1^{K[3]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right ) \int _1^{K[3]}\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]dK[3]+\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] \left (\int _1^xc \exp \left (\int _1^{K[4]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[4]+c_2\right )+c_1\right ) \end{align*}

✗ Sympy

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

NotImplementedError : solve: Cannot solve b*y(x)/(x**2*(-a + x)**2) - c + Derivative(y(x), (x, 2))