problem Problem 5(d)

Internal problem ID [12250]
Internal ﬁle name [OUTPUT/10903_Thursday_September_28_2023_01_08_26_AM_73359447/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(d).
ODE order: 2.
ODE degree: 1.

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

2.30.1 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) &=2 x\\ q(x) &=\frac {\sin \left (x \right )}{x}\\ F &=\frac {\sinh \left (x \right )}{x} \end {align*}

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

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)] 
trying symmetries linear in x and y(x) 
-> Try solving first the homogeneous part of the ODE 
   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 
      -> 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 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 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 
   <- unable to find a useful change of variables 
      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 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 
         -> 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 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 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 
      <- unable to find a useful change of variables 
         trying a symmetry of the form [xi=0, eta=F(x)] 
      trying to convert to an ODE of Bessel type 
      -> trying reduction of order to Riccati 
         trying Riccati sub-methods: 
            trying Riccati_symmetries 
            -> 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 with_periodic_functions in the coefficients 
         --- Trying Lie symmetry methods, 2nd order --- 
         `, `-> Computing symmetries using: way = 5`[0, y]

2.30 problem Problem 5(d)

2.30.1 Existence and uniqueness analysis