Internal problem ID [3365]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 22
Problem number: 625.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Bernoulli]
Solve \begin {gather*} \boxed {3 y^{2} y^{\prime }-1-x -a y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 154
dsolve(3*y(x)^2*diff(y(x),x) = 1+x+a*y(x)^3,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left ({\mathrm e}^{a x} c_{1} a^{2}-a x -a -1\right ) a \right )^{\frac {1}{3}}}{a} \\ y \relax (x ) = -\frac {\left (\left ({\mathrm e}^{a x} c_{1} a^{2}-a x -a -1\right ) a \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (\left ({\mathrm e}^{a x} c_{1} a^{2}-a x -a -1\right ) a \right )^{\frac {1}{3}}}{2 a} \\ y \relax (x ) = -\frac {\left (\left ({\mathrm e}^{a x} c_{1} a^{2}-a x -a -1\right ) a \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (\left ({\mathrm e}^{a x} c_{1} a^{2}-a x -a -1\right ) a \right )^{\frac {1}{3}}}{2 a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.518 (sec). Leaf size: 111
DSolve[3 y[x]^2 y'[x]==1+x+a y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{a^2 c_1 e^{a x}-a (x+1)-1}}{a^{2/3}} \\ y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{a^2 c_1 e^{a x}-a (x+1)-1}}{a^{2/3}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{a^2 c_1 e^{a x}-a (x+1)-1}}{a^{2/3}} \\ \end{align*}