Internal problem ID [2923]
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]
Solve \begin {gather*} \boxed {y^{\prime } x -a \,x^{3} \left (-y x +1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 126
dsolve(x*diff(y(x),x) = a*x^3*(1-x*y(x))*y(x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {9 \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{\frac {1}{3}} \left (-9 a \,x^{3}\right )^{\frac {1}{3}}}{2 \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} 3^{\frac {5}{6}} x \pi \left (-9 a \,x^{3}\right )^{\frac {1}{3}}-9 \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} x \Gamma \left (\frac {1}{3}, -\frac {a \,x^{3}}{3}\right ) \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{\frac {1}{3}}-9 \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} c_{1} \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{\frac {1}{3}} \left (-9 a \,x^{3}\right )^{\frac {1}{3}}-9 x \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{\frac {1}{3}} \left (-9 a \,x^{3}\right )^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.225 (sec). Leaf size: 47
DSolve[x y'[x]==a x^3(1-x y[x])y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {3 e^{\frac {a x^3}{3}}}{-a x^4 E_{-\frac {1}{3}}\left (-\frac {a x^3}{3}\right )+3 c_1} \\ y(x)\to 0 \\ \end{align*}