55.33.24 problem 233

Internal problem ID [14007]
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 : 233
Date solved : Thursday, October 02, 2025 at 09:08:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 143

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

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

✓ Mathematica. Time used: 2.094 (sec). Leaf size: 152

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

\begin{align*} y(x)&\to e^{-\frac {(a+b+\lambda ) (\log (x-a)-\log (x-b))}{a-b}} \left (c_1 \exp \left (\frac {\left (\sqrt {\mu } \sqrt {\frac {(a+b+\lambda )^2}{\mu }-4}+a+b+\lambda \right ) (\log (x-a)-\log (x-b))}{2 (a-b)}\right )+c_2 \exp \left (\frac {\left (-\sqrt {\mu } \sqrt {\frac {(a+b+\lambda )^2}{\mu }-4}+a+b+\lambda \right ) (\log (x-a)-\log (x-b))}{2 (a-b)}\right )\right ) \end{align*}

✗ Sympy

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

False