2.1.6 Problem 16 (a)

Internal problem ID [18465]
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 : Monday, March 31, 2025 at 05:30:07 PM
CAS classification : [[_2nd_order, _missing_x]]

✗ Maple

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

$$\text{No solution found}$$

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 sing\ 
ular 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 metho\ 
ds 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 gener\ 
al case 
-> trying 2nd order, dynamical_symmetries, only a reduction of order through on\ 
e integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 
--- 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
 

✗ Mathematica

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

Not solved

✗ Sympy

from sympy import * 
u = symbols("u") 
v = Function("v") 
ode = Eq(-(Derivative(v(u), u)**4 + 1/v(u))**(1/3) + Derivative(v(u), (u, 2)),0) 
ics = {} 
dsolve(ode,func=v(u),ics=ics)
 

NotImplementedError : The given ODE -(Derivative(v(u), (u, 2))**3 - 1/v(u))**(1/4) + Derivative(v(u), u) cannot be solved by the factorable group method