61.34.14 problem 14

Internal problem ID [12699]
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 : 14
Date solved : Monday, March 31, 2025 at 06:51:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 36

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

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

✓ Mathematica. Time used: 1.227 (sec). Leaf size: 44

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

\[ y(x)\to e^{-\frac {a e^{\lambda x}}{\lambda }-1} \left (c_2 \operatorname {ExpIntegralEi}\left (\frac {(a-b) e^{x \lambda }}{\lambda }\right )+e c_1\right ) \]

✗ Sympy

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

False