Internal problem ID [11113]
Internal file name [OUTPUT/10370_Wednesday_January_24_2024_10_18_19_PM_28952538/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: 26.
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.
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +b c -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y=0} \]
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*c*t^2-c^2*t^2+a*k*t+b*c*t-2*c*k*t+b*k+c*t-k^2)*u(t)+(a*t^2+b*t+t)*diff(u(t),t)+t^2*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.391 (sec). Leaf size: 114
dsolve(diff(y(x),x$2)+(a*exp(x)+b)*diff(y(x),x)+( c*(a-c)*exp(2*x)+ (a*k+b*c+c-2*c*k)*exp(x) + k*(b-k) )*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\operatorname {WhittakerM}\left (-\frac {b}{2}+k , -\frac {b}{2}+k +\frac {1}{2}, \left (-2 c +a \right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {a \,{\mathrm e}^{x}}{2}-\frac {b x}{2}} \left (-2 c +a \right )^{-b +2 k} c_{2} +\left (\left (-2 c +a \right ) {\mathrm e}^{x}\right )^{-\frac {b}{2}+k} c_{2} \left (-2 c +a \right )^{-b +2 k} \left (-1+b -2 k \right ) {\mathrm e}^{\left (-a +c \right ) {\mathrm e}^{x}-\frac {b x}{2}}+c_{1} {\mathrm e}^{-k x -{\mathrm e}^{x} c} \]
✓ Solution by Mathematica
Time used: 3.806 (sec). Leaf size: 71
DSolve[y''[x]+(a*Exp[x]+b)*y'[x]+( c*(a-c)*Exp[2*x]+ (a*k+b*c+c-2*c*k)*Exp[x] + k*(b-k) )*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-c e^x} \left (e^x\right )^{-k} \left (c_1-c_2 \left (e^x\right )^{2 k-b} \left (e^x (a-2 c)\right )^{b-2 k} \Gamma \left (2 k-b,(a-2 c) e^x\right )\right ) \]