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 : Monday, January 27, 2025 at 02:52:07 PM
CAS classification : [_separable]

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

✓ Solution by Maple

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

dsolve((x+2*x^3)+(y(x)+2*y(x)^3)*diff(y(x),x)=0,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*}

✓ Solution by Mathematica

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

DSolve[(x+2*x^3)+(y[x]+2*y[x]^3)*D[y[x],x]==0,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*}