Internal problem ID [11099]
Internal file name [OUTPUT/10356_Wednesday_January_24_2024_10_18_13_PM_71465684/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: 12.
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.
\[ \boxed {y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y=0} \]
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 symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> 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 0F1 ODE <- Whittaker successful <- special function solution successful Change of variables used: [x = ln(t)/lambda] Linear ODE actually solved: (4*a*t^2*(b*t+c)^n-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: 0.796 (sec). Leaf size: 224
dsolve(diff(y(x),x$2)-diff(y(x),x)+(a*exp(2*lambda*x)*(b*exp(lambda*x)+c)^n+1/4-1/4*lambda^2 )*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\Gamma \left (\frac {n +1}{n +2}\right )^{2} {\mathrm e}^{-\frac {x \left (\lambda -1\right )}{2}} {\left (-\frac {a \left (b \,{\mathrm e}^{x \lambda }+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}\right )}^{\frac {1}{2 n +4}} c_{1} \left (n +2\right ) \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \left (b \,{\mathrm e}^{x \lambda }+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}}\right )+\csc \left (\frac {\pi \left (n +1\right )}{n +2}\right ) {\left (-\frac {a \left (b \,{\mathrm e}^{x \lambda }+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}\right )}^{-\frac {1}{2 n +4}} \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \left (b \,{\mathrm e}^{x \lambda }+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}}\right ) \pi c_{2} \left (b \,{\mathrm e}^{\frac {x \left (1+\lambda \right )}{2}}+{\mathrm e}^{-\frac {x \left (\lambda -1\right )}{2}} c \right )}{\left (n +2\right ) \Gamma \left (\frac {n +1}{n +2}\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]-y'[x]+(a*Exp[2*\[Lambda]*x]*(b*Exp[\[Lambda]*x]+c)^n+1/4-1/4*\[Lambda]^2 )*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved