31.1.20 problem 10.2

Internal problem ID [5718]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 2
Problem number : 10.2
Date solved : Monday, January 27, 2025 at 01:11:24 PM
CAS classification : [_rational, _Bernoulli]

\begin{align*} 3 z^{2} z^{\prime }-a z^{3}&=1+x \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 104

dsolve(3*z(x)^2*diff(z(x),x)-a*z(x)^3=x+1,z(x), singsol=all)
 

\begin{align*} z \left (x \right ) &= \frac {{\left (a \left (c_{1} {\mathrm e}^{a x} a^{2}-1+\left (-x -1\right ) a \right )\right )}^{{1}/{3}}}{a} \\ z \left (x \right ) &= -\frac {{\left (a \left (c_{1} {\mathrm e}^{a x} a^{2}-1+\left (-x -1\right ) a \right )\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 a} \\ z \left (x \right ) &= \frac {{\left (a \left (c_{1} {\mathrm e}^{a x} a^{2}-1+\left (-x -1\right ) a \right )\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 23.876 (sec). Leaf size: 111

DSolve[3*z[x]^2*D[z[x],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*}