61.34.24 problem 24

Internal problem ID [12709]
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 : 24
Date solved : Monday, March 31, 2025 at 06:51:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 54

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

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

✓ Mathematica. Time used: 0.335 (sec). Leaf size: 82

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

\[ y(x)\to \frac {\left (c_2 e^{x \sqrt {b^2-4 c}}+c_1 \sqrt {b^2-4 c}\right ) e^{-\frac {a e^{\lambda x}}{\lambda }-\frac {1}{2} x \left (\sqrt {b^2-4 c}+b\right )}}{\sqrt {b^2-4 c}} \]

✗ Sympy

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

False