61.32.23 problem 232

Internal problem ID [12654]
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 : 232
Date solved : Monday, March 31, 2025 at 06:49:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -a \right )^{2} \left (x -b \right )^{2} y^{\prime \prime }-c y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 104

ode:=(x-a)^2*(x-b)^2*diff(diff(y(x),x),x)-c*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.54 (sec). Leaf size: 168

ode=(x-a)^2*(x-b)^2*D[y[x],{x,2}]-c*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

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