66.1.49 problem Problem 63

Internal problem ID [13896]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 63
Date solved : Tuesday, January 28, 2025 at 06:07:52 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 71

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

\begin{align*} y &= \frac {{\left (\left (x^{2}+c_{1} \right ) x^{2}\right )}^{{1}/{3}}}{x} \\ y &= -\frac {{\left (\left (x^{2}+c_{1} \right ) x^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 x} \\ y &= \frac {{\left (\left (x^{2}+c_{1} \right ) x^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.199 (sec). Leaf size: 72

DSolve[3*x*y[x]^2*D[y[x],x]+y[x]^3-2*x==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{x^2+c_1}}{\sqrt [3]{x}} \\ y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{x^2+c_1}}{\sqrt [3]{x}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{x^2+c_1}}{\sqrt [3]{x}} \\ \end{align*}