61.34.10 problem 10

Internal problem ID [12774]
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
Date solved : Tuesday, January 28, 2025 at 04:19:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+a y^{\prime }+\left (b \,{\mathrm e}^{\lambda x}+c \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.964 (sec). Leaf size: 69

dsolve(diff(y(x),x$2)+a*diff(y(x),x)+(b*exp(lambda*x)+c)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {a x}{2}} \left (\operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-4 c}}{\lambda }, \frac {2 \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {a^{2}-4 c}}{\lambda }, \frac {2 \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.106 (sec). Leaf size: 123

DSolve[D[y[x],{x,2}]+a*D[y[x],x]+(b*Exp[\[Lambda]*x]+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-\frac {a x}{2}} \left (c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^2-4 c}}{\lambda }\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^2-4 c}}{\lambda },\frac {2 \sqrt {b e^{x \lambda }}}{\lambda }\right )+c_2 \operatorname {Gamma}\left (\frac {\lambda +\sqrt {a^2-4 c}}{\lambda }\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^2-4 c}}{\lambda },\frac {2 \sqrt {b e^{x \lambda }}}{\lambda }\right )\right ) \]