Internal problem ID [3866]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 2
Problem number: 10.2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Bernoulli]
Solve \begin {gather*} \boxed {3 z^{2} z^{\prime }-a z^{3}-x -1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 154
dsolve(3*z(x)^2*diff(z(x),x)-a*z(x)^3=x+1,z(x), singsol=all)
\begin{align*} z \relax (x ) = \frac {\left (\left ({\mathrm e}^{a x} c_{1} a^{2}-a x -a -1\right ) a \right )^{\frac {1}{3}}}{a} \\ z \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} \\ z \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: 1.818 (sec). Leaf size: 111
DSolve[3*z[x]^2*z'[x]-a*z[x]^3==x+1,z[x],x,IncludeSingularSolutions -> True]
\begin{align*} z(x)\to \frac {\sqrt [3]{a^2 c_1 e^{a x}-a (x+1)-1}}{a^{2/3}} \\ z(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{a^2 c_1 e^{a x}-a (x+1)-1}}{a^{2/3}} \\ z(x)\to \frac {(-1)^{2/3} \sqrt [3]{a^2 c_1 e^{a x}-a (x+1)-1}}{a^{2/3}} \\ \end{align*}