54.3.76 problem 1090

Internal problem ID [12371]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1090
Date solved : Wednesday, October 01, 2025 at 01:44:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a^{2} y^{\prime \prime }+a \left (a^{2}-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: 40

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

\[ y = {\mathrm e}^{-\frac {x \,a^{3}+2 b \,{\mathrm e}^{-a x}}{2 a^{2}}} \left (c_1 \sinh \left (\frac {a x}{2}\right )+c_2 \cosh \left (\frac {a x}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 45

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

\begin{align*} y(x)&\to \frac {e^{-\frac {b e^{-a x}}{a^2}-a x} \left (a^2 c_1 e^{a x}-b c_2\right )}{a^2} \end{align*}

✗ Sympy

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

False