Internal problem ID [11107]
Internal file name [OUTPUT/10364_Wednesday_January_24_2024_10_18_17_PM_66837224/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: 20.
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.
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+c \left (a \,{\mathrm e}^{\lambda x}+b -c \right ) y=0} \]
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)/lambda] Linear ODE actually solved: (a*c*t+b*c-c^2)*u(t)+(a*lambda*t^2+b*lambda*t+lambda^2*t)*diff(u(t),t)+lambda^2*t^2*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 176
dsolve(diff(y(x),x$2)+(a*exp(lambda*x)+b)*diff(y(x),x)+c*(a*exp(lambda*x)+b-c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\frac {-{\mathrm e}^{x \lambda } a -\left (b +3 \lambda \right ) x \lambda }{2 \lambda }} c_{2} \left (-\lambda -2 c +b \right )^{2} \operatorname {WhittakerM}\left (-\frac {-\lambda -2 c +b}{2 \lambda }, -\frac {-2 \lambda -2 c +b}{2 \lambda }, \frac {a \,{\mathrm e}^{x \lambda }}{\lambda }\right )+\left (\left (\lambda +2 c -b \right ) {\mathrm e}^{\frac {-{\mathrm e}^{x \lambda } a -\left (b +3 \lambda \right ) x \lambda }{2 \lambda }}+a \,{\mathrm e}^{\frac {-{\mathrm e}^{x \lambda } a -x \lambda \left (b +\lambda \right )}{2 \lambda }}\right ) c_{2} \lambda \operatorname {WhittakerM}\left (-\frac {b -2 c +\lambda }{2 \lambda }, -\frac {-2 \lambda -2 c +b}{2 \lambda }, \frac {a \,{\mathrm e}^{x \lambda }}{\lambda }\right )+c_{1} {\mathrm e}^{-c x} \]
✓ Solution by Mathematica
Time used: 0.244 (sec). Leaf size: 96
DSolve[y''[x]+(a*Exp[\[Lambda]*x]+b)*y'[x]+c*(a*Exp[\[Lambda]*x]+b-c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-1)^{-\frac {c}{\lambda }} c^{c/\lambda } \lambda ^{\frac {c}{\lambda }-1} a^{-\frac {c}{\lambda }} \left (c e^{\lambda x}\right )^{-\frac {c}{\lambda }} \left (c_2 (2 c-b) (-1)^{c/\lambda } \Gamma \left (-\frac {b-2 c}{\lambda },0,\frac {a e^{x \lambda }}{\lambda }\right )+c_1 \lambda \right ) \]