Internal problem ID [11112]
Internal file name [OUTPUT/10369_Wednesday_January_24_2024_10_18_18_PM_66780811/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: 25.
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.
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b -\lambda \right ) y^{\prime }+\left (c \,{\mathrm e}^{2 \lambda x}+{\mathrm e}^{\lambda x} a b +b^{2}-b \lambda \right ) y=0} \]
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 Group is reducible or imprimitive <- Kovacics algorithm successful Change of variables used: [x = ln(t)/lambda] Linear ODE actually solved: (a*b*t+c*t^2+b^2-b*lambda)*u(t)+(a*lambda*t^2+2*b*lambda*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: 0.484 (sec). Leaf size: 74
dsolve(diff(y(x),x$2)+(a*exp(lambda*x)+2*b-lambda)*diff(y(x),x)+(c*exp(2*lambda*x)+a*b*exp(lambda*x)+b^2-b*lambda)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {-2 b \lambda x +{\mathrm e}^{x \lambda } \sqrt {a^{2}-4 c}-{\mathrm e}^{x \lambda } a}{2 \lambda }}+c_{2} {\mathrm e}^{-\frac {2 b \lambda x +{\mathrm e}^{x \lambda } \sqrt {a^{2}-4 c}+{\mathrm e}^{x \lambda } a}{2 \lambda }} \]
✓ Solution by Mathematica
Time used: 2.119 (sec). Leaf size: 97
DSolve[y''[x]+(a*Exp[\[Lambda]*x]+2*b-\[Lambda])*y'[x]+(c*Exp[2*\[Lambda]*x]+a*b*Exp[\[Lambda]*x]+b^2-b*\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\left (e^{\lambda x}\right )^{-\frac {b}{\lambda }} e^{-\frac {\left (\sqrt {a^2-4 c}+a\right ) e^{\lambda x}}{2 \lambda }} \left (c_2 \lambda e^{\frac {\sqrt {a^2-4 c} e^{\lambda x}}{\lambda }}+c_1 \sqrt {a^2-4 c}\right )}{\sqrt {a^2-4 c}} \]