61.34.3 problem 3

Internal problem ID [12688]
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 : 3
Date solved : Monday, March 31, 2025 at 06:51:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 32

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

\[ y = {\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} \left (c_2 \,\operatorname {Ei}_{1}\left (-\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }\right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.888 (sec). Leaf size: 51

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

\[ y(x)\to e^{-\frac {a e^{\lambda x}}{\lambda }} \left (c_2 \int _1^{e^{x \lambda }}\frac {e^{\frac {2 a K[1]}{\lambda }}}{K[1]}dK[1]+c_1\right ) \]

✗ Sympy

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

False