problem 2.4 (iii)

Internal problem ID [12569]
Internal ﬁle name [OUTPUT/11222_Wednesday_October_18_2023_10_01_21_PM_93984434/index.tex]

Book: Nonlinear Ordinary Diﬀerential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section: Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number: 2.4 (iii).
ODE order: 2.
ODE degree: 1.

1.16.1 Solving as second order ode missing x ode

This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(x\) an independent variable. Using \begin {align*} x' &= p(x) \end {align*}

Then \begin {align*} x'' &= \frac {dp}{dt}\\ &= \frac {dx}{dt} \frac {dp}{dx}\\ &= p \frac {dp}{dx} \end {align*}

Hence the ode becomes \begin {align*} p \left (x \right ) \left (\frac {d}{d x}p \left (x \right )\right )+p \left (x \right )+\left (-x^{2}+1\right ) x = 0 \end {align*}

Which is now solved as ﬁrst order ode for \(p(x)\). Unable to determine ODE type.

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
trying 2nd order WeierstrassP 
trying 2nd order JacobiSN 
differential order: 2; trying a linearization to 3rd order 
trying 2nd order ODE linearizable_by_differentiation 
trying 2nd order, 2 integrating factors of the form mu(x,y) 
trying differential order: 2; missing variables 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = exp_sym 
-> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)+_b(_a)+_a-_a^3 = 0, _b(_a)`   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying 1st order linear 
   trying Bernoulli 
   trying separable 
   trying inverse linear 
   trying homogeneous types: 
   trying Chini 
   differential order: 1; looking for linear symmetries 
   trying exact 
   trying Abel 
   Looking for potential symmetries 
   Looking for potential symmetries 
   Looking for potential symmetries 
   trying inverse_Riccati 
   trying an equivalence to an Abel ODE 
   differential order: 1; trying a linearization to 2nd order 
   --- trying a change of variables {x -> y(x), y(x) -> x} 
   differential order: 1; trying a linearization to 2nd order 
   trying 1st order ODE linearizable_by_differentiation 
   --- Trying Lie symmetry methods, 1st order --- 
   `, `-> Computing symmetries using: way = 3 
   `, `-> Computing symmetries using: way = 4 
   `, `-> Computing symmetries using: way = 2 
   trying symmetry patterns for 1st order ODEs 
   -> 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 symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
   -> trying a symmetry pattern of the form [F(x),G(x)] 
   -> trying a symmetry pattern of the form [F(y),G(y)] 
   -> 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 a symmetry pattern of conformal type 
-> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 
   --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- 
   -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) 
trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases 
trying differential order: 2; exact nonlinear 
trying 2nd order, integrating factor of the form mu(x,y) 
-> trying 2nd order, the S-function method 
   -> trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for the S-function 
   -> trying 2nd order, the S-function method 
   -> trying 2nd order, No Point Symmetries Class V 
      --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- 
      -> trying 2nd order, No Point Symmetries Class V 
   -> trying 2nd order, No Point Symmetries Class V 
      --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- 
      -> trying 2nd order, No Point Symmetries Class V 
   -> trying 2nd order, No Point Symmetries Class V 
      --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- 
      -> trying 2nd order, No Point Symmetries Class V 
trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case 
-> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^ 
   --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- 
   -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)* 
--- Trying Lie symmetry methods, 2nd order --- 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = 5 
`, `-> Computing symmetries using: way = formal 
            *** Sublevel 2 *** 
            Methods for first order ODEs: 
            --- Trying classification methods --- 
            trying a quadrature 
            trying 1st order linear 
            <- 1st order linear successful 
   -> Calling odsolve with the ODE`, diff(y(x), x), y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful`

1.16 problem 2.4 (iii)

1.16.1 Solving as second order ode missing x ode