Internal problem ID [8597]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1017.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left ({\mathrm e}^{2 x}-v^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 17
dsolve(diff(diff(y(x),x),x)+(exp(2*x)-v^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \BesselJ \left (v , {\mathrm e}^{x}\right )+c_{2} \BesselY \left (v , {\mathrm e}^{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 46
DSolve[(E^(2*x) - v^2)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \text {Gamma}(1-v) J_{-v}\left (\sqrt {e^{2 x}}\right )+c_2 \text {Gamma}(v+1) J_v\left (\sqrt {e^{2 x}}\right ) \\ \end{align*}