Internal problem ID [11114]
Internal file name [OUTPUT/10371_Wednesday_January_24_2024_10_18_19_PM_37319352/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: 27.
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.
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y=0} \]
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 -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Whittaker successful <- special function solution successful Change of variables used: [x = ln(t)/lambda] Linear ODE actually solved: (alpha*t^2+beta*t+gamma)*u(t)+(a*lambda*t^2+b*lambda*t+lambda^2*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: 0.422 (sec). Leaf size: 141
dsolve(diff(y(x),x$2)+(a*exp(lambda*x)+b)*diff(y(x),x)+( alpha*exp(2*lambda*x)+ beta*exp(lambda*x) + gamma )*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\frac {-{\mathrm e}^{x \lambda } a -x \lambda \left (b +\lambda \right )}{2 \lambda }} \left (\operatorname {WhittakerM}\left (-\frac {a \left (b +\lambda \right )-2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, \lambda }, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {a \left (b +\lambda \right )-2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, \lambda }, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 2.375 (sec). Leaf size: 248
DSolve[y''[x]+(a*Exp[\[Lambda]*x]+b)*y'[x]+( alpha*Exp[2*\[Lambda]*x]+ \[Beta]*Exp[\[Lambda]*x] + \[Gamma] )*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {\left (\sqrt {a^2-4 \alpha }+a\right ) e^{\lambda x}}{2 \lambda }} \left (e^{\lambda x}\right )^{\frac {\sqrt {b^2-4 \gamma }-b}{2 \lambda }} \left (c_1 \operatorname {HypergeometricU}\left (\frac {-2 \beta +a (b+\lambda )+\sqrt {a^2-4 \alpha } \left (\lambda +\sqrt {b^2-4 \gamma }\right )}{2 \sqrt {a^2-4 \alpha } \lambda },\frac {\lambda +\sqrt {b^2-4 \gamma }}{\lambda },\frac {\sqrt {a^2-4 \alpha } e^{x \lambda }}{\lambda }\right )+c_2 L_{\frac {2 \beta -a (b+\lambda )-\sqrt {a^2-4 \alpha } \left (\lambda +\sqrt {b^2-4 \gamma }\right )}{2 \sqrt {a^2-4 \alpha } \lambda }}^{\frac {\sqrt {b^2-4 \gamma }}{\lambda }}\left (\frac {\sqrt {a^2-4 \alpha } e^{x \lambda }}{\lambda }\right )\right ) \]