2.1.8 Problem 17 (a)

Internal problem ID [18467]
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 : 17 (a)
Date solved : Monday, March 31, 2025 at 05:30:14 PM
CAS classification : [NONE]

✗ Maple

ode:=(diff(y(x),x)+y(x))^(1/2) = (diff(diff(y(x),x),x)+2*x)^(1/4); 
dsolve(ode,y(x), 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 
-> 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 meth\ 
ods for dynamical symmetries --- 
   -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through o\ 
ne 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 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 gener\ 
al 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 metho\ 
ds 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 m\ 
ethods 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 m\ 
ethods 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 m\ 
ethods 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 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 a change of variables {x -> y(x), y(x) -> x} and re-entering meth\ 
ods 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)*y)^2 
solving 2nd order ODE of high degree, Lie methods 
   -> Computing symmetries using: way = 3 
   -> Computing symmetries using: way = 5
 

✗ Mathematica

ode=Sqrt[D[y[x],x]+y[x]]== (D[y[x],{x,2}]+2*x)^(1/4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(2*x + Derivative(y(x), (x, 2))) + y(x) + Derivative(y(x), x) cannot be solved by the factorable group method