2.29 problem Problem 5(c)

Internal problem ID [12249]
Internal file name [OUTPUT/10902_Thursday_September_28_2023_01_08_26_AM_88437475/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(c).
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.

\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+\left (x -1\right ) y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}

2.29.1 Existence and uniqueness analysis

This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime \prime } + p(x)y^{\prime } + q(x) y &= F \end {align*}

Where here \begin {align*} p(x) &=\frac {x -1}{x^{2}+1}\\ q(x) &=\frac {1}{x^{2}+1}\\ F &=0 \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }+\frac {\left (x -1\right ) y^{\prime }}{x^{2}+1}+\frac {y}{x^{2}+1} = 0 \end {align*}

The domain of \(p(x)=\frac {x -1}{x^{2}+1}\) is \[ \{-\infty

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
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 
      <- heuristic approach successful 
   <- hypergeometric successful 
<- special function solution successful`
 

✓ Solution by Maple

Time used: 0.328 (sec). Leaf size: 157

dsolve([(x^2+1)*diff(y(x),x$2)+(x-1)*diff(y(x),x)+y(x)=0,y(0) = 0, D(y)(0) = 1],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {-20 \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}-\frac {i}{2}\right ], \frac {1}{2}\right ) {\mathrm e}^{\left (\frac {1}{4}-\frac {i}{4}\right ) \pi } \left (x +i\right )^{\frac {1}{2}+\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i}{2}, \frac {1}{2}+\frac {3 i}{2}\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )+20 \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i}{2}, \frac {1}{2}+\frac {3 i}{2}\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )}{\left (10-10 i\right ) \left (\operatorname {hypergeom}\left (\left [1-i, 1+i\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}\right )-\operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}-\frac {i}{2}\right ], \frac {1}{2}\right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i}{2}, \frac {1}{2}+\frac {3 i}{2}\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}\right )+\left (-1+7 i\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}+\frac {3 i}{2}, \frac {3}{2}-\frac {i}{2}\right ], \left [\frac {5}{2}+\frac {i}{2}\right ], \frac {1}{2}\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}-\frac {i}{2}\right ], \frac {1}{2}\right )} \]

✗ Solution by Mathematica

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

DSolve[{(x^2+1)*y''[x]+(x-1)*y'[x]+y[x]==0,{y[0]==0,y'[0]==1}},y[x],x,IncludeSingularSolutions -> True]
 

Not solved