Internal problem ID [8670]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1090.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {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 {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 59
dsolve(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,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {a^{3} x +2 b \,{\mathrm e}^{-a x}}{2 a^{2}}} \sinh \left (\frac {a x}{2}\right )+c_{2} {\mathrm e}^{-\frac {a^{3} x +2 b \,{\mathrm e}^{-a x}}{2 a^{2}}} \cosh \left (\frac {a x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 45
DSolve[(b^2*y[x])/E^(2*a*x) + a*(a^2 - (2*b)/E^(a*x))*y'[x] + a^2*y''[x] == 0,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*}