Internal problem ID [11118]
Internal file name [OUTPUT/10375_Wednesday_January_24_2024_10_18_20_PM_62185204/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: 31.
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.
\[ \boxed {y^{\prime \prime }+{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{2 \mu x}+b \right ) y^{\prime }+\mu \left ({\mathrm e}^{\lambda x} \left (b -a \,{\mathrm e}^{2 \mu x}\right )-\mu \right ) y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: trying Riccati_symmetries -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[0, y]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 81
dsolve(diff(y(x),x$2)+exp(lambda*x)*(a*exp(2*mu*x)+b)*diff(y(x),x)+mu*(exp(lambda*x)*(b-a*exp(2*mu*x))-mu)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\left (\int \frac {{\mathrm e}^{\frac {-{\mathrm e}^{x \left (\lambda +2 \mu \right )} a \lambda -2 \left (\frac {\lambda }{2}+\mu \right ) \left (-2 \lambda \mu x +b \,{\mathrm e}^{x \lambda }\right )}{\lambda \left (\lambda +2 \mu \right )}}}{\left ({\mathrm e}^{2 x \mu } a +b \right )^{2}}d x \right ) c_{2} +c_{1} \right ) \left (a \,{\mathrm e}^{x \mu }+{\mathrm e}^{-x \mu } b \right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]+Exp[\[Lambda]*x]*(a*Exp[2*\[Mu]*x]+b)*y'[x]+\[Mu]*(Exp[\[Lambda]*x]*(b-a*Exp[2*\[Mu]*x])-\[Mu])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved