55.28.20 problem 30

Internal problem ID [13803]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 30
Date solved : Thursday, October 02, 2025 at 08:06:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 46

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

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

✓ Mathematica. Time used: 0.1 (sec). Leaf size: 86

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

\begin{align*} y(x)&\to \frac {e^{-\frac {1}{2} x \left (2 \sqrt {a+b^2-c}+a x+2 b\right )} \left (c_2 e^{2 x \sqrt {a+b^2-c}}+2 c_1 \sqrt {a+b^2-c}\right )}{2 \sqrt {a+b^2-c}} \end{align*}

✗ Sympy

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

False