29.6.29 problem 175

Internal problem ID [4775]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 6
Problem number : 175
Date solved : Monday, January 27, 2025 at 09:37:19 AM
CAS classification : [_Bernoulli]

\begin{align*} x y^{\prime }&=a \,x^{3} \left (1-y x \right ) y \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 95

dsolve(x*diff(y(x),x) = a*x^3*(1-x*y(x))*y(x),y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {3 \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{{1}/{3}} 3^{{2}/{3}}}{-3 \,3^{{2}/{3}} \Gamma \left (\frac {2}{3}\right ) {\mathrm e}^{-\frac {a \,x^{3}}{3}} c_{1} \left (-a \,x^{3}\right )^{{1}/{3}}-3 \,3^{{2}/{3}} \Gamma \left (\frac {2}{3}\right ) x \left (-a \,x^{3}\right )^{{1}/{3}}+2 \sqrt {3}\, \pi \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} x -3 \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} \Gamma \left (\frac {2}{3}\right ) \Gamma \left (\frac {1}{3}, -\frac {a \,x^{3}}{3}\right ) x} \]

✓ Solution by Mathematica

Time used: 0.232 (sec). Leaf size: 66

DSolve[x D[y[x],x]==a x^3(1-x y[x])y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {e^{\frac {a x^3}{3}} \sqrt [3]{-a x^3}}{\sqrt [3]{3} x \Gamma \left (\frac {4}{3},-\frac {a x^3}{3}\right )+c_1 \sqrt [3]{-a x^3}} \\ y(x)\to 0 \\ \end{align*}