60.3.28 problem 1033

Internal problem ID [11024]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1033
Date solved : Sunday, March 30, 2025 at 07:39:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

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

\[ y = c_1 \sin \left ({\mathrm e}^{-x} \sqrt {a}\right )+c_2 \cos \left ({\mathrm e}^{-x} \sqrt {a}\right ) \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 37

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

\[ y(x)\to c_1 \cos \left (\sqrt {a} e^{-x}\right )-c_2 \sin \left (\sqrt {a} e^{-x}\right ) \]

✗ Sympy

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

False