Internal problem ID [11103]
Internal file name [OUTPUT/10360_Wednesday_January_24_2024_10_18_15_PM_73998394/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: 16.
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.
\[ \boxed {y^{\prime \prime }+2 k \,{\mathrm e}^{\mu x} y^{\prime }+\left ({\mathrm e}^{2 \lambda x} a +b \,{\mathrm e}^{\lambda x}+k^{2} {\mathrm e}^{2 \mu x}+k \mu \,{\mathrm e}^{\mu x}+c \right ) y=0} \]
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*k*exp(mu*x)*diff(y(x),x)+(a*exp(2*lambda*x)+b*exp(lambda*x)+k^2*exp(2*mu*x)+k*mu*exp(mu*x)+c)*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 2.531 (sec). Leaf size: 232
DSolve[y''[x]+2*k*Exp[\[Mu]*x]*y'[x]+(a*Exp[2*\[Lambda]*x]+b*Exp[\[Lambda]*x]+k^2*Exp[2*\[Mu]*x]+k*\[Mu]*Exp[\[Mu]*x]+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {1}{2}-\frac {i \sqrt {c}}{\lambda }} \left (e^x\right )^{\frac {1}{2}-\frac {\lambda }{2}} \left (\left (e^x\right )^{\mu }\right )^{\left .-\frac {1}{2}\right /\mu } \left (\left (e^x\right )^{\lambda }\right )^{\frac {1}{2}-\frac {i \sqrt {c}}{\lambda }} e^{-\frac {k \left (e^x\right )^{\mu }}{\mu }+\frac {i \sqrt {a} \left (e^x\right )^{\lambda }}{\lambda }} \left (c_1 \operatorname {HypergeometricU}\left (-\frac {\frac {i b}{\sqrt {a}}-\lambda +2 i \sqrt {c}}{2 \lambda },1-\frac {2 i \sqrt {c}}{\lambda },-\frac {2 i \sqrt {a} \left (e^x\right )^{\lambda }}{\lambda }\right )+c_2 L_{\frac {\frac {i b}{\sqrt {a}}-\lambda +2 i \sqrt {c}}{2 \lambda }}^{-\frac {2 i \sqrt {c}}{\lambda }}\left (-\frac {2 i \sqrt {a} \left (e^x\right )^{\lambda }}{\lambda }\right )\right ) \]