61.34.21 problem 21

Internal problem ID [12706]
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 : 21
Date solved : Monday, March 31, 2025 at 06:51:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.235 (sec). Leaf size: 248

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

\[ y(x)\to \frac {-\frac {1}{2} c_2 (a-2 \lambda ) e^{-\frac {b e^{2 \lambda x}}{2 \lambda }} \left (b \lambda e^{2 \lambda x}\right )^{-\frac {a}{2 \lambda }} \left (b 2^{\frac {a}{2 \lambda }} \lambda ^{a/\lambda } e^{2 \lambda x}+\operatorname {Gamma}\left (1-\frac {a}{2 \lambda }\right ) e^{\frac {b e^{2 \lambda x}}{2 \lambda }} \left (a+b e^{2 \lambda x}\right ) \left (b \lambda e^{2 \lambda x}\right )^{\frac {a}{2 \lambda }}-e^{\frac {b e^{2 \lambda x}}{2 \lambda }} \left (a+b e^{2 \lambda x}\right ) \left (b \lambda e^{2 \lambda x}\right )^{\frac {a}{2 \lambda }} \Gamma \left (1-\frac {a}{2 \lambda },\frac {b e^{2 x \lambda }}{2 \lambda }\right )\right )-\frac {2 i c_1 \lambda ^2 \left (a+b e^{2 \lambda x}\right )}{a}}{\sqrt {2} \lambda \sqrt {b \lambda e^{2 \lambda x}}} \]

✗ Sympy

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

False