55.35.8 problem 8

Internal problem ID [14045]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.3-1. Equations with exponential functions
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:09:53 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} b \,{\mathrm e}^{2 a x} y+a y^{\prime }+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 78

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

\begin{align*} y(x)&\to \frac {\sqrt {a} e^{-\frac {a x}{2}} \left (2 c_1 \cos \left (\frac {\sqrt {b e^{2 a x}}}{a}\right )+c_2 \sin \left (\frac {\sqrt {b e^{2 a x}}}{a}\right )\right )}{\sqrt {2} \sqrt [4]{b e^{2 a x}}} \end{align*}

✗ Sympy

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

False