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 : Wednesday, March 05, 2025 at 08:17:31 PM
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: 0.469 (sec). Leaf size: 53

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);
 

\[ y = c_{1} {\mathrm e}^{\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\lambda x}}{2 \lambda }}+c_{2} {\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\lambda x}}{2 \lambda }} \]

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

False