60.3.32 problem 1033

Internal problem ID [11042]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1033
Date solved : Monday, January 27, 2025 at 10:42:24 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*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 27

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

\[ y = c_{1} \sin \left ({\mathrm e}^{-x} \sqrt {a}\right )+c_{2} \cos \left ({\mathrm e}^{-x} \sqrt {a}\right ) \]

✓ Solution by Mathematica

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

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

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