61.34.19 problem 19

Internal problem ID [12704]
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 : 19
Date solved : Monday, March 31, 2025 at 06:51:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 48

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

\[ y = \left (c_1 \,{\mathrm e}^{\frac {{\mathrm e}^{\lambda x} \sqrt {a^{2}-4 b}}{\lambda }}+c_2 \right ) {\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\lambda x}}{2 \lambda }} \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 61

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

\[ y(x)\to e^{-\frac {\left (\sqrt {a^2-4 b}+a\right ) e^{\lambda x}}{2 \lambda }} \left (c_2 e^{\frac {\sqrt {a^2-4 b} e^{\lambda x}}{\lambda }}+c_1\right ) \]

✗ Sympy

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

False