47.1.16 problem 16

Internal problem ID [7397]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 16
Date solved : Wednesday, March 05, 2025 at 04:25:29 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 113

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

\begin{align*} y &= -\frac {\sqrt {-2-2 \sqrt {-4 x^{4}-4 x^{2}-8 c_{1} -1}}}{2} \\ y &= \frac {\sqrt {-2-2 \sqrt {-4 x^{4}-4 x^{2}-8 c_{1} -1}}}{2} \\ y &= -\frac {\sqrt {-2+2 \sqrt {-4 x^{4}-4 x^{2}-8 c_{1} -1}}}{2} \\ y &= \frac {\sqrt {-2+2 \sqrt {-4 x^{4}-4 x^{2}-8 c_{1} -1}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 2.163 (sec). Leaf size: 151

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

\begin{align*} y(x)\to -\frac {\sqrt {-1-\sqrt {-4 x^4-4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-1-\sqrt {-4 x^4-4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-1+\sqrt {-4 x^4-4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-1+\sqrt {-4 x^4-4 x^2+1+8 c_1}}}{\sqrt {2}} \\ \end{align*}

✓ Sympy. Time used: 3.399 (sec). Leaf size: 122

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- \sqrt {C_{1} - 4 x^{4} - 4 x^{2}} - 1}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- \sqrt {C_{1} - 4 x^{4} - 4 x^{2}} - 1}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {\sqrt {C_{1} - 4 x^{4} - 4 x^{2}} - 1}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {\sqrt {C_{1} - 4 x^{4} - 4 x^{2}} - 1}}{2}\right ] \]