Internal problem ID [3431]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 6
Problem number: 175.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {x y^{\prime }-a \,x^{3} \left (-y x +1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 95
dsolve(x*diff(y(x),x) = a*x^3*(1-x*y(x))*y(x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {3 \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{\frac {1}{3}} 3^{\frac {2}{3}}}{-3 \Gamma \left (\frac {2}{3}\right ) {\mathrm e}^{-\frac {a \,x^{3}}{3}} 3^{\frac {2}{3}} c_{1} \left (-a \,x^{3}\right )^{\frac {1}{3}}-3 \Gamma \left (\frac {2}{3}\right ) 3^{\frac {2}{3}} x \left (-a \,x^{3}\right )^{\frac {1}{3}}+2 \pi \sqrt {3}\, {\mathrm e}^{-\frac {a \,x^{3}}{3}} x -3 \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} \Gamma \left (\frac {2}{3}\right ) \Gamma \left (\frac {1}{3}, -\frac {a \,x^{3}}{3}\right ) x} \]
✓ Solution by Mathematica
Time used: 0.224 (sec). Leaf size: 66
DSolve[x y'[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*}