Internal problem ID [11091]
Internal file name [OUTPUT/10348_Wednesday_January_24_2024_10_18_11_PM_34915829/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: 4.
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.
\[ \boxed {y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y=0} \]
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 <- Kovacics algorithm successful Change of variables used: [x = ln(t)] Linear ODE actually solved: (-a^2*t^2-2*a*b*t-a*t-b^2)*u(t)+t*diff(u(t),t)+t^2*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 76
dsolve(diff(y(x),x$2)-(a^2*exp(2*x)+a*(2*b+1)*exp(x)+b^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -c_{2} a^{-2 b} \operatorname {WhittakerM}\left (-b , \frac {1}{2}-b , 2 a \,{\mathrm e}^{x}\right )+c_{1} {\mathrm e}^{b x +a \,{\mathrm e}^{x}}+\left (a \,{\mathrm e}^{x}\right )^{-b} {\mathrm e}^{-a \,{\mathrm e}^{x}} c_{2} a^{-2 b} \left (b 2^{-b +1}-2^{-b}\right ) \]
✓ Solution by Mathematica
Time used: 1.8 (sec). Leaf size: 57
DSolve[y''[x]-(a^2*Exp[2*x]+a*(2*b+1)*Exp[x]+b^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{a e^x} \left (e^x\right )^{-b} \left (c_1 \left (e^x\right )^{2 b}-4^b c_2 \left (a e^x\right )^{2 b} \Gamma \left (-2 b,2 a e^x\right )\right ) \]