Internal problem ID [11101]
Internal file name [OUTPUT/10358_Wednesday_January_24_2024_10_18_14_PM_71707269/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: 14.
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.
\[ \boxed {y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y=0} \]
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 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*b*t+a*lambda)*u(t)+(a*lambda*t+b*lambda*t+lambda^2)*diff(u(t),t)+lambda^2*t*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 36
dsolve(diff(y(x),x$2)+(a+b)*exp(lambda*x)*diff(y(x),x)+a*exp(lambda*x)*(b*exp(lambda*x)+lambda)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {a \,{\mathrm e}^{x \lambda }}{\lambda }} \left (c_{1} +\operatorname {expIntegral}_{1}\left (-\frac {{\mathrm e}^{x \lambda } \left (a -b \right )}{\lambda }\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 2.377 (sec). Leaf size: 40
DSolve[y''[x]+(a+b)*Exp[\[Lambda]*x]*y'[x]+a*Exp[\[Lambda]*x]*(b*Exp[\[Lambda]*x]+\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {a e^{\lambda x}}{\lambda }} \left (c_2 \operatorname {ExpIntegralEi}\left (\frac {(a-b) e^{x \lambda }}{\lambda }\right )+c_1\right ) \]