15.2.22 problem 22

Internal problem ID [2892]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 6, page 25
Problem number : 22
Date solved : Sunday, March 30, 2025 at 12:46:11 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y^{2} \left (y y^{\prime }-x \right )+x^{3}&=0 \end{align*}

✓ Maple. Time used: 0.147 (sec). Leaf size: 56

ode:=y(x)^2*(y(x)*diff(y(x),x)-x)+x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (\sqrt {3}\, \ln \left (x^{4} \sec \left (\textit {\_Z} \right )^{2}\right )+\sqrt {3}\, \ln \left (3\right )-2 \sqrt {3}\, \ln \left (2\right )+4 \sqrt {3}\, c_1 -2 \textit {\_Z} \right )\right )-1\right ) x \]

✓ Mathematica. Time used: 0.118 (sec). Leaf size: 63

ode=y[x]^2*(y[x]*D[y[x],x]-x)+x^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {\arctan \left (\frac {\frac {2 y(x)^2}{x^2}-1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (\frac {y(x)^4}{x^4}-\frac {y(x)^2}{x^2}+1\right )=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 4.239 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3 + (-x + y(x)*Derivative(y(x), x))*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \log {\left (x \right )} = C_{1} - \log {\left (\sqrt [4]{1 - \frac {y^{2}{\left (x \right )}}{x^{2}} + \frac {y^{4}{\left (x \right )}}{x^{4}}} \right )} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \left (1 - \frac {2 y^{2}{\left (x \right )}}{x^{2}}\right )}{3} \right )}}{6} \]