Internal problem ID [11108]
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: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+a y^{\prime }+\left (b \,{\mathrm e}^{\lambda x}+c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 73
dsolve(diff(y(x),x$2)+a*diff(y(x),x)+(b*exp(lambda*x)+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {a x}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-4 c}}{\lambda }, \frac {2 \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )+c_{2} {\mathrm e}^{-\frac {a x}{2}} \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-4 c}}{\lambda }, \frac {2 \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right ) \]
✓ Solution by Mathematica
Time used: 0.185 (sec). Leaf size: 123
DSolve[y''[x]+a*y'[x]+(b*Exp[\[Lambda]*x]+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {a x}{2}} \left (c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^2-4 c}}{\lambda }\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^2-4 c}}{\lambda },\frac {2 \sqrt {b e^{x \lambda }}}{\lambda }\right )+c_2 \operatorname {Gamma}\left (\frac {\lambda +\sqrt {a^2-4 c}}{\lambda }\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^2-4 c}}{\lambda },\frac {2 \sqrt {b e^{x \lambda }}}{\lambda }\right )\right ) \]