Internal problem ID [11110]
Internal file name [OUTPUT/10367_Wednesday_January_24_2024_10_18_18_PM_71646621/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: 23.
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.
\[ \boxed {y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left ({\mathrm e}^{2 \lambda x} a^{2}+{\mathrm e}^{\mu x} c \right ) y=0} \]
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
dsolve(diff(y(x),x$2)+(2*a*exp(lambda*x)-lambda)*diff(y(x),x)+(a^2*exp(2*lambda*x)+c*exp(mu*x))*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 1.858 (sec). Leaf size: 164
DSolve[y''[x]+(2*a*Exp[\[Lambda]*x]-\[Lambda])*y'[x]+(a^2*Exp[2*\[Lambda]*x]+c*Exp[\[Mu]*x])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-1)^{-\frac {\lambda }{\mu }} 2^{\frac {\lambda +\mu }{2 \mu }} \left (\left (e^x\right )^{\lambda }\right )^{\frac {\lambda -1}{2 \lambda }} \left (e^x\right )^{\frac {1}{2}-\frac {\mu }{2}} e^{-\frac {a \left (e^x\right )^{\lambda }}{\lambda }} \left (\left (e^x\right )^{\mu }\right )^{\frac {\lambda +\mu }{2 \mu }} \left (-\frac {c \left (e^x\right )^{\mu }}{\mu ^2}\right )^{-\frac {\lambda }{2 \mu }} \left (c_1 (-1)^{\lambda /\mu } \operatorname {BesselI}\left (\frac {\lambda }{\mu },2 \sqrt {-\frac {c \left (e^x\right )^{\mu }}{\mu ^2}}\right )+c_2 K_{\frac {\lambda }{\mu }}\left (2 \sqrt {-\frac {c \left (e^x\right )^{\mu }}{\mu ^2}}\right )\right ) \]