Internal problem ID [11098]
Internal file name [OUTPUT/10355_Wednesday_January_24_2024_10_18_13_PM_46931920/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: 11.
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.
\[ \boxed {y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y=0} \]
Maple trace
`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 <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful Change of variables used: [x = ln(t)/lambda] Linear ODE actually solved: (4*a*t^3+4*b*t^2-lambda^2+1)*u(t)+(4*lambda^2*t-4*lambda*t)*diff(u(t),t)+4*lambda^2*t^2*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 51
dsolve(diff(y(x),x$2)-diff(y(x),x)+(a*exp(3*lambda*x)+b*exp(2*lambda*x)+1/4-1/4*lambda^2 )*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (\lambda -1\right )}{2}} \left (\operatorname {AiryAi}\left (-\frac {{\mathrm e}^{x \lambda } a +b}{\lambda ^{\frac {2}{3}} a^{\frac {2}{3}}}\right ) c_{1} +\operatorname {AiryBi}\left (-\frac {{\mathrm e}^{x \lambda } a +b}{\lambda ^{\frac {2}{3}} a^{\frac {2}{3}}}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 1.332 (sec). Leaf size: 77
DSolve[y''[x]-y'[x]+(a*Exp[3*\[Lambda]*x]+b*Exp[2*\[Lambda]*x]+1/4-1/4*\[Lambda]^2 )*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{x/2} \left (c_1 \operatorname {AiryAi}\left (\frac {\left (e^{x \lambda } a+b\right ) \sqrt [3]{-\frac {a}{\lambda ^2}}}{a}\right )+c_2 \operatorname {AiryBi}\left (\frac {\left (e^{x \lambda } a+b\right ) \sqrt [3]{-\frac {a}{\lambda ^2}}}{a}\right )\right )}{\sqrt {e^{\lambda x}}} \]