Internal problem ID [8613]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1033.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime }+a \,{\mathrm e}^{-2 x} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (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)
\[ y \relax (x ) = c_{1} \sin \left ({\mathrm e}^{-x} \sqrt {a}\right )+c_{2} \cos \left ({\mathrm e}^{-x} \sqrt {a}\right ) \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 37
DSolve[(a*y[x])/E^(2*x) + y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cos \left (\sqrt {a} e^{-x}\right )-c_2 \sin \left (\sqrt {a} e^{-x}\right ) \\ \end{align*}