Internal problem ID [11089]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.3-1. Equations with
exponential functions
Problem number: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+{\mathrm e}^{\lambda x} y a=0} \]
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 39
dsolve(diff(y(x),x$2)+a*exp(lambda*x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (0, \frac {2 \sqrt {a}\, {\mathrm e}^{\frac {x \lambda }{2}}}{\lambda }\right )+c_{2} \operatorname {BesselY}\left (0, \frac {2 \sqrt {a}\, {\mathrm e}^{\frac {x \lambda }{2}}}{\lambda }\right ) \]
✓ Solution by Mathematica
Time used: 0.062 (sec). Leaf size: 55
DSolve[y''[x]+a*Exp[\[Lambda]*x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {e^{x \lambda }}}{\lambda }\right )+2 c_2 \operatorname {BesselY}\left (0,\frac {2 \sqrt {a} \sqrt {e^{x \lambda }}}{\lambda }\right ) \]