55.35.13 problem 13

Internal problem ID [14050]
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 : 13
Date solved : Thursday, October 02, 2025 at 09:09:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 a \,{\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{\lambda x}+\lambda \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to (c_2 x+c_1) e^{-\frac {a e^{\lambda x}}{\lambda }} \end{align*}

✗ Sympy

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

False