29.9.7 problem 247

Internal problem ID [4847]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 9
Problem number : 247
Date solved : Monday, January 27, 2025 at 09:42:34 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 113

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

\begin{align*} y \left (x \right ) &= -\frac {3^{{1}/{3}} \left (x^{2} \left (x^{3}-3 c_{1} \right )^{2}\right )^{{1}/{3}}}{x^{3}-3 c_{1}} \\ y \left (x \right ) &= \frac {\left (x^{2} \left (x^{3}-3 c_{1} \right )^{2}\right )^{{1}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )}{2 x^{3}-6 c_{1}} \\ y \left (x \right ) &= \frac {\left (x^{2} \left (x^{3}-3 c_{1} \right )^{2}\right )^{{1}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )}{-2 x^{3}+6 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.200 (sec). Leaf size: 89

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

\begin{align*} y(x)\to -\frac {\sqrt [3]{-3} x^{2/3}}{\sqrt [3]{-x^3+3 c_1}} \\ y(x)\to \frac {x^{2/3}}{\sqrt [3]{-\frac {x^3}{3}+c_1}} \\ y(x)\to \frac {(-1)^{2/3} x^{2/3}}{\sqrt [3]{-\frac {x^3}{3}+c_1}} \\ y(x)\to 0 \\ \end{align*}