55.35.38 problem 38

Internal problem ID [14075]
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 : 38
Date solved : Friday, October 03, 2025 at 07:24:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 133

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

\[ y = c_1 \,\operatorname {csgn}\left (a \right ) \sin \left (k \sqrt {-\frac {1}{a^{2} {\mathrm e}^{2 \lambda x}+b}}\, \sqrt {a^{2} {\mathrm e}^{2 \lambda x}+b}\, \ln \left (\left (\sqrt {a^{2} {\mathrm e}^{2 \lambda x}+b}\, \operatorname {csgn}\left (a \right )+{\mathrm e}^{\lambda x} a \right ) \operatorname {csgn}\left (a \right )\right )\right )+c_2 \cos \left (k \sqrt {-\frac {1}{a^{2} {\mathrm e}^{2 \lambda x}+b}}\, \sqrt {a^{2} {\mathrm e}^{2 \lambda x}+b}\, \ln \left (\left (\sqrt {a^{2} {\mathrm e}^{2 \lambda x}+b}\, \operatorname {csgn}\left (a \right )+{\mathrm e}^{\lambda x} a \right ) \operatorname {csgn}\left (a \right )\right ) \operatorname {csgn}\left (a \right )\right ) \]

✓ Mathematica. Time used: 0.266 (sec). Leaf size: 69

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

\begin{align*} y(x)&\to c_1 \cosh \left (k \text {arctanh}\left (\frac {a e^{\lambda x}}{\sqrt {a^2 e^{2 \lambda x}+b}}\right )\right )+i c_2 \sinh \left (k \text {arctanh}\left (\frac {a e^{\lambda x}}{\sqrt {a^2 e^{2 \lambda x}+b}}\right )\right ) \end{align*}

✗ Sympy

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

False