Internal problem ID [11090]
Internal file name [OUTPUT/10347_Wednesday_January_24_2024_10_18_11_PM_29877762/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: 3.
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.
\[ \boxed {y^{\prime \prime }+a \left (\lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{2 \lambda x}\right ) y=0} \]
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^2*t+a*lambda)*u(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: 32
dsolve(diff(y(x),x$2)+a*(lambda*exp(lambda*x)-a*exp(2*lambda*x))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {a \,{\mathrm e}^{x \lambda }}{\lambda }} \left (c_{1} +\operatorname {expIntegral}_{1}\left (-\frac {2 a \,{\mathrm e}^{x \lambda }}{\lambda }\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 1.32 (sec). Leaf size: 37
DSolve[y''[x]+a*(\[Lambda]*Exp[\[Lambda]*x]-a*Exp[2*\[Lambda]*x])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {a e^{\lambda x}}{\lambda }} \left (c_2 \operatorname {ExpIntegralEi}\left (\frac {2 a e^{x \lambda }}{\lambda }\right )+c_1\right ) \]