34.5 problem 5

Internal problem ID [11092]
Internal file name [OUTPUT/10349_Wednesday_January_24_2024_10_18_11_PM_29171069/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: 5.
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.

\[ \boxed {y^{\prime \prime }-\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+c \right ) y=0} \]

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: 
      (-a*t^2-b*t-c)*u(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: 0.343 (sec). Leaf size: 73

dsolve(diff(y(x),x$2)-(a*exp(2*lambda*x)+b*exp(lambda*x)+c)*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{-\frac {x \lambda }{2}} \left (\operatorname {WhittakerM}\left (-\frac {b}{2 \lambda \sqrt {a}}, \frac {\sqrt {c}}{\lambda }, \frac {2 \sqrt {a}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {b}{2 \lambda \sqrt {a}}, \frac {\sqrt {c}}{\lambda }, \frac {2 \sqrt {a}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 1.158 (sec). Leaf size: 145

DSolve[y''[x]-(a*Exp[2*\[Lambda]*x]+b*Exp[\[Lambda]*x]+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-\frac {\sqrt {a} e^{\lambda x}}{\lambda }} \left (e^{\lambda x}\right )^{\frac {\sqrt {c}}{\lambda }} \left (c_1 \operatorname {HypergeometricU}\left (\frac {\frac {b}{\sqrt {a}}+\lambda +2 \sqrt {c}}{2 \lambda },\frac {2 \sqrt {c}}{\lambda }+1,\frac {2 \sqrt {a} e^{x \lambda }}{\lambda }\right )+c_2 L_{-\frac {\frac {b}{\sqrt {a}}+\lambda +2 \sqrt {c}}{2 \lambda }}^{\frac {2 \sqrt {c}}{\lambda }}\left (\frac {2 \sqrt {a} e^{x \lambda }}{\lambda }\right )\right ) \]