Internal problem ID [4236]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter IX, Special forms of differential equations. Examples XVII. page 247
Problem number: 18.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+{\mathrm e}^{2 x} y-n^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 17
dsolve(diff(y(x),x$2)+exp(2*x)*y(x)=n^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \BesselJ \left (n , {\mathrm e}^{x}\right )+c_{2} \BesselY \left (n , {\mathrm e}^{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 46
DSolve[y''[x]+Exp[2*x]*y[x]==n^2*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \text {Gamma}(1-n) \text {BesselJ}\left (-n,\sqrt {e^{2 x}}\right )+c_2 \text {Gamma}(n+1) \text {BesselJ}\left (n,\sqrt {e^{2 x}}\right ) \\ \end{align*}