2.1.6 problem 16 (a)

Internal problem ID [18212]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter 1. section 5. Problems at page 19
Problem number : 16 (a)
Date solved : Friday, December 20, 2024 at 05:57:06 AM
CAS classification : [[_2nd_order, _missing_x]]

Solve

\begin{align*} v^{\prime \prime }&=\left (\frac {1}{v}+{v^{\prime }}^{4}\right )^{{1}/{3}} \end{align*}

Maple step by step solution

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)^4*_a+1)/_a)^(1/3) = 0, _b(_a)`   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying homogeneous types: 
   differential order: 1; looking for linear symmetries 
   trying exact 
   Looking for potential symmetries 
   trying an equivalence to an Abel ODE 
   trying 1st order ODE linearizable_by_differentiation 
   --- Trying Lie symmetry methods, 1st order --- 
   `, `-> Computing symmetries using: way = 2 
   `, `-> Computing symmetries using: way = 3 
   `, `-> Computing symmetries using: way = 4 
   `, `-> Computing symmetries using: way = 5 
   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 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 2nd order, No Point Symmetries Class V 
   -> 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 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`
 

Maple dsolve solution

Solving time : 0.140 (sec)
Leaf size : maple_leaf_size

dsolve(diff(diff(v(u),u),u) = (1/v(u)+diff(v(u),u)^4)^(1/3), 
       v(u),singsol=all)
 

\[ \text {No solution found} \]

Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

DSolve[{D[v[u],{u,2}]==(1/v[u]+D[v[u],u]^4)^(1/3),{}}, 
       v[u],u,IncludeSingularSolutions->True]
 

Not solved