61.34.18 problem 18

Internal problem ID [12703]
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 : 18
Date solved : Monday, March 31, 2025 at 06:51:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.262 (sec). Leaf size: 79

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

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

✓ Mathematica. Time used: 0.121 (sec). Leaf size: 129

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

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

✗ Sympy

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

False