Internal problem ID [11102]
Internal file name [OUTPUT/10359_Wednesday_January_24_2024_10_18_14_PM_13126995/index.tex]
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: 15.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{\mu x} \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+\mu \right ) y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 46
dsolve(diff(y(x),x$2)+a*exp(lambda*x)*diff(y(x),x)-b*exp(mu*x)*(a*exp(lambda*x)+b*exp(mu*x)+mu)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\left (\int {\mathrm e}^{\frac {-2 b \,{\mathrm e}^{x \mu } \lambda -{\mathrm e}^{x \lambda } a \mu }{\mu \lambda }}d x \right ) c_{1} +c_{2} \right ) {\mathrm e}^{\frac {b \,{\mathrm e}^{x \mu }}{\mu }} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]+a*Exp[\[Lambda]*x]*y'[x]-b*Exp[\[Mu]*x]*(a*Exp[\[Lambda]*x]+b*Exp[\[Mu]*x]+\[Mu])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved