problem 24

Internal problem ID [710]
Internal ﬁle name [OUTPUT/710_Sunday_June_05_2022_01_47_31_AM_72098390/index.tex]

Book: Elementary diﬀerential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section: Chapter 3, Second order linear equations, 3.7 Forced Vibrations. page 217
Problem number: 24.
ODE order: 2.
ODE degree: 1.

\[ \boxed {u^{\prime \prime }+u^{\prime }+\frac {u^{3}}{5}=\cos \left (t \right )} \] With initial conditions \begin {align*} [u \left (0\right ) = 2, u^{\prime }\left (0\right ) = 0] \end {align*}

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 
-> 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 symmetries linear in x and y(x) 
trying differential order: 2; exact nonlinear 
trying 2nd order, integrating factor of the form mu(y) 
trying 2nd order, integrating factor of the form mu(x,y) 
trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case 
trying 2nd order, integrating factor of the form mu(y,y) 
trying differential order: 2; mu polynomial in y 
trying 2nd order, integrating factor of the form mu(x,y) 
differential order: 2; looking for linear symmetries 
-> 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`

12.4 problem 24