Internal problem ID [11105]
Internal file name [OUTPUT/10362_Wednesday_January_24_2024_10_18_16_PM_88334329/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: 18.
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.
\[ \boxed {y^{\prime \prime }+\left ({\mathrm e}^{2 \lambda x} a +\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y=0} \]
Maple trace Kovacic algorithm successful
`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 trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible Solution has integrals. Trying a special function solution free of integrals... -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre <- Kummer successful <- special function solution successful -> Trying to convert hypergeometric functions to elementary form... <- elementary form is not straightforward to achieve - returning special function solution free of uncomputed integrals <- Kovacics algorithm successful Change of variables used: [x = ln(t)/lambda] Linear ODE actually solved: -t*a*u(t)+(a*t^2+2*lambda)*diff(u(t),t)+t*lambda*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.593 (sec). Leaf size: 79
dsolve(diff(y(x),x$2)+(a*exp(2*lambda*x)+lambda)*diff(y(x),x)-a*lambda*exp(2*lambda*x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{2} \sqrt {\pi }\, \left ({\mathrm e}^{x \lambda } a +{\mathrm e}^{-x \lambda } \lambda \right ) \operatorname {erf}\left (\frac {\sqrt {2}\, {\mathrm e}^{x \lambda } \sqrt {a}}{2 \sqrt {\lambda }}\right )+\sqrt {a}\, \sqrt {\lambda }\, {\mathrm e}^{-\frac {a \,{\mathrm e}^{2 x \lambda }}{2 \lambda }} \sqrt {2}\, c_{2} +c_{1} \left ({\mathrm e}^{x \lambda } a +{\mathrm e}^{-x \lambda } \lambda \right ) \]
✓ Solution by Mathematica
Time used: 0.283 (sec). Leaf size: 129
DSolve[y''[x]+(a*Exp[2*\[Lambda]*x]+\[Lambda])*y'[x]-a*\[Lambda]*Exp[2*\[Lambda]*x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\sqrt {2 \pi } c_2 \left (a e^{2 \lambda x}+\lambda \right ) \text {erf}\left (\frac {\sqrt {a \lambda e^{2 \lambda x}}}{\sqrt {2} \lambda }\right )-4 i \sqrt {2} a c_1 e^{2 \lambda x}+2 c_2 e^{-\frac {a e^{2 \lambda x}}{2 \lambda }} \sqrt {a \lambda e^{2 \lambda x}}-4 i \sqrt {2} c_1 \lambda }{4 \sqrt {a \lambda e^{2 \lambda x}}} \]