34.6.14 problem 18

Internal problem ID [6088]
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
Date solved : Monday, January 27, 2025 at 01:35:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+{\mathrm e}^{2 x} y&=n^{2} y \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 17

dsolve(diff(y(x),x$2)+exp(2*x)*y(x)=n^2*y(x),y(x), singsol=all)
 

\[ y = c_1 \operatorname {BesselJ}\left (n , {\mathrm e}^{x}\right )+c_2 \operatorname {BesselY}\left (n , {\mathrm e}^{x}\right ) \]

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 46

DSolve[D[y[x],{x,2}]+Exp[2*x]*y[x]==n^2*y[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {Gamma}(1-n) \operatorname {BesselJ}\left (-n,\sqrt {e^{2 x}}\right )+c_2 \operatorname {Gamma}(n+1) \operatorname {BesselJ}\left (n,\sqrt {e^{2 x}}\right ) \]