61.34.17 problem 17

Internal problem ID [12702]
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 : 17
Date solved : Monday, March 31, 2025 at 06:51:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 35

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

\[ y(x)\to \frac {e^{\frac {b e^{a x}}{a}} \left (b c_2 e^{a x}+a c_1\right )}{a} \]

✗ Sympy

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

False