89.31.3 problem 3

Internal problem ID [24948]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 246
Problem number : 3
Date solved : Thursday, October 02, 2025 at 11:36:28 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} {y^{\prime }}^{2}+x y^{2} y^{\prime }+y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.208 (sec). Leaf size: 124

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

\begin{align*} y &= \frac {4}{x^{2}} \\ y &= 0 \\ y &= \frac {2 \sqrt {2}\, x -2 c_1}{c_1 \left (c_1^{2}-2 x^{2}\right )} \\ y &= \frac {-2 \sqrt {2}\, x -2 c_1}{c_1 \left (c_1^{2}-2 x^{2}\right )} \\ y &= -\frac {\left (\sqrt {2}\, x c_1 -2\right ) c_1^{2}}{2 c_1^{2} x^{2}-4} \\ y &= \frac {\left (\sqrt {2}\, x c_1 +2\right ) c_1^{2}}{2 c_1^{2} x^{2}-4} \\ \end{align*}

✓ Mathematica. Time used: 0.521 (sec). Leaf size: 152

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

\begin{align*} \text {Solve}\left [-\frac {\sqrt {y(x)} \sqrt {4-x^2 y(x)} \text {arcsinh}\left (\frac {1}{2} x \sqrt {-y(x)}\right )}{\sqrt {-y(x)} \sqrt {x^2 y(x)-4}}-\frac {1}{2} \log (y(x))=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {\sqrt {y(x)} \sqrt {4-x^2 y(x)} \text {arcsinh}\left (\frac {1}{2} x \sqrt {-y(x)}\right )}{\sqrt {-y(x)} \sqrt {x^2 y(x)-4}}-\frac {1}{2} \log (y(x))=c_1,y(x)\right ]\\ y(x)&\to 0\\ y(x)&\to \frac {4}{x^2} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE x*y(x)**2/2 - sqrt((x**2*y(x) - 4)*y(x)**3)/2 + Derivative(y(x),