2.28 problem Problem 5(b)

Internal problem ID [12248]
Internal file name [OUTPUT/10901_Thursday_September_28_2023_01_08_25_AM_74340670/index.tex]

Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number: Problem 5(b).
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.

\[ \boxed {y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cot \left (x \right ) y=0} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{4}\right ) = 1, y^{\prime }\left (\frac {\pi }{4}\right ) = 0\right ] \end {align*}

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 a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      -> Whittaker 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      -> hypergeometric 
         -> heuristic approach 
         -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
      -> Mathieu 
         -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   -> 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 a symmetry of the form [xi=0, eta=F(x)] 
      trying 2nd order exact linear 
      trying symmetries linear in x and y(x) 
      trying to convert to a linear ODE with constant coefficients 
      -> trying with_periodic_functions in the coefficients 
         --- Trying Lie symmetry methods, 2nd order --- 
         `, `-> Computing symmetries using: way = 5 
   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 
      -> hypergeometric 
         -> heuristic approach 
         -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
               <- hyper3 successful: received ODE is equivalent to the 2F1 ODE 
      <- hypergeometric successful 
   <- special function solution successful 
   Change of variables used: 
      [x = arccot(t)] 
   Linear ODE actually solved: 
      t^2*u(t)+(2*t^4+t^2-1)*diff(u(t),t)+(t^5+2*t^3+t)*diff(diff(u(t),t),t) = 0 
<- change of variables successful`
 

✓ Solution by Maple

Time used: 6.454 (sec). Leaf size: 46435

dsolve([diff(y(x),x$2)+tan(x)*diff(y(x),x)+cot(x)*y(x)=0,y(1/4*Pi) = 1, D(y)(1/4*Pi) = 0],y(x), singsol=all)
 

\[ \text {Expression too large to display} \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[{y''[x]+Tan[x]*y'[x]+Cot[x]*y[x]==0,{y[Pi/4]==1,y'[Pi/4]==0}},y[x],x,IncludeSingularSolutions -> True]
 

Not solved