Internal problem ID [11109]
Internal file name [OUTPUT/10366_Wednesday_January_24_2024_10_18_17_PM_14375704/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: 22.
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.
\[ \boxed {y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (b -\lambda \right ) {\mathrm e}^{2 \lambda x} y=0} \]
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 -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Kummer successful <- special function solution successful Change of variables used: [x = ln(t)/lambda] Linear ODE actually solved: (a^2*b*t-a^2*lambda*t)*u(t)+(b*t+a+b-2*lambda)*diff(u(t),t)+t*lambda*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.406 (sec). Leaf size: 205
dsolve(diff(y(x),x$2)+(a+b*exp(lambda*x)+b-3*lambda)*diff(y(x),x)+a^2*lambda*(b-lambda)*exp(2*lambda*x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {{\mathrm e}^{x \lambda } \left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right )}{2 \lambda }} \left (\operatorname {KummerU}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (-2 \lambda +b +a \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {-2 \lambda +b +a}{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{2} +\operatorname {KummerM}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (-2 \lambda +b +a \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {-2 \lambda +b +a}{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 3.799 (sec). Leaf size: 260
DSolve[y''[x]+(a+b*Exp[\[Lambda]*x]+b-3*\[Lambda])*y'[x]+a^2*\[Lambda]*(b-\[Lambda])*Exp[2*\[Lambda]*x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \exp \left (-\frac {e^{\lambda x} \left (\sqrt {-4 a^2 b \lambda +4 a^2 \lambda ^2+b^2}+b\right )}{2 \lambda }\right ) \left (c_1 \operatorname {HypergeometricU}\left (\frac {(a+b-2 \lambda ) \left (b+\sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}\right )}{2 \lambda \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}},\frac {a+b-2 \lambda }{\lambda },\frac {e^{x \lambda } \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}{\lambda }\right )+c_2 L_{-\frac {(a+b-2 \lambda ) \left (b+\sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}\right )}{2 \lambda \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}}^{\frac {a+b-3 \lambda }{\lambda }}\left (\frac {e^{x \lambda } \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}{\lambda }\right )\right ) \]