55.33.7 problem 217
Internal
problem
ID
[13990]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-7
Problem
number
:
217
Date
solved
:
Thursday, October 02, 2025 at 09:08:27 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y&=c \,x^{2} \left (x -a \right )^{2} \end{align*}
✓ Maple. Time used: 0.115 (sec). Leaf size: 219
ode:=x^2*(x-a)^2*diff(diff(y(x),x),x)+b*y(x) = c*x^2*(x-a)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (-\int \sqrt {x \left (-x +a \right )}\, \left (\frac {-x +a}{x}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x \left (\frac {-x +a}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c +\int \sqrt {x \left (-x +a \right )}\, \left (\frac {x}{-x +a}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x \left (\frac {x}{-x +a}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c +\left (\frac {x}{-x +a}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_1 \sqrt {a^{2}-4 b}+\left (\frac {-x +a}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_2 \sqrt {a^{2}-4 b}\right ) \sqrt {x \left (-x +a \right )}}{\sqrt {a^{2}-4 b}}
\]
✓ Mathematica. Time used: 35.591 (sec). Leaf size: 371
ode=x^2*(x-a)^2*D[y[x],{x,2}]+b*y[x]==c*x^2*(x-a)^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {a c x^2 (a-x) \left (1-\frac {x}{a}\right )^{-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}-\frac {1}{2}} \left (\left (\sqrt {1-\frac {4 b}{a^2}}-3\right ) \left (1-\frac {x}{a}\right )^{\sqrt {1-\frac {4 b}{a^2}}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}-\frac {1}{2},\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {3}{2},\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {5}{2},\frac {x}{a}\right )+\left (\sqrt {1-\frac {4 b}{a^2}}+3\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}-\frac {1}{2},\frac {3}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}},\frac {5}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}},\frac {x}{a}\right )\right )}{2 \left (2 a^2+b\right ) \sqrt {1-\frac {4 b}{a^2}}}+c_1 x^{\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {1}{2}} (x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}}+\frac {c_2 x^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} (x-a)^{\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {1}{2}}}{a \sqrt {1-\frac {4 b}{a^2}}} \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) - c*x**2*(-a + x)**2 + x**2*(-a + x)**2*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve b*y(x) - c*x**2*(-a + x)**2 + x**2*(-a + x)**2*Derivative(y(x), (x, 2))