problem 32

Internal problem ID [1186]
Internal ﬁle name [OUTPUT/1187_Sunday_June_05_2022_02_04_30_AM_23388380/index.tex]

Book: Elementary diﬀerential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number: 32.
ODE order: 2.
ODE degree: 1.

\[ \boxed {\left (x -1\right )^{2} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+\left (x -1\right )^{3} y=\left (x -1\right )^{3} {\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = -6] \end {align*}

10.32.1 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^{2}+1}{\left (x -1\right )^{2}}\\ q(x) &=x -1\\ F &=\left (x -1\right ) {\mathrm e}^{x} \end {align*}

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

The domain of \(p(x)=\frac {-x^{2}+1}{\left (x -1\right )^{2}}\) is \[ \{x <1\boldsymbol {\lor }1

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
-> Try solving first the homogeneous part of the ODE 
   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 
      -> Mathieu 
         -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius 
   trying a solution in terms of MeijerG functions 
   -> 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 symmetries linear in x and y(x) 
      trying to convert to a linear ODE with constant coefficients 
      trying 2nd order, integrating factor of the form mu(x,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 
         trying 2nd order exact linear 
         trying symmetries linear in x and y(x) 
         trying to convert to a linear ODE with constant coefficients 
         trying to convert to an ODE of Bessel type 
      trying to convert to an ODE of Bessel type 
      -> trying reduction of order to Riccati 
         trying Riccati sub-methods: 
            -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
            -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
            -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
--- Trying Lie symmetry methods, 2nd order --- 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = 5`

10.32 problem 32

10.32.1 Existence and uniqueness analysis