25.1.20 problem 10.2
Internal
problem
ID
[6853]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
2
Problem
number
:
10.2
Date
solved
:
Tuesday, September 30, 2025 at 03:55:53 PM
CAS
classification
:
[_rational, _Bernoulli]
\begin{align*} 3 z^{2} z^{\prime }-a z^{3}&=x +1 \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 104
ode:=3*z(x)^2*diff(z(x),x)-a*z(x)^3 = 1+x;
dsolve(ode,z(x), singsol=all);
\begin{align*}
z &= \frac {{\left (a \left ({\mathrm e}^{a x} c_1 \,a^{2}-1+a \left (-x -1\right )\right )\right )}^{{1}/{3}}}{a} \\
z &= -\frac {{\left (a \left ({\mathrm e}^{a x} c_1 \,a^{2}-1+a \left (-x -1\right )\right )\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 a} \\
z &= \frac {{\left (a \left ({\mathrm e}^{a x} c_1 \,a^{2}-1+a \left (-x -1\right )\right )\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 a} \\
\end{align*}
✓ Mathematica. Time used: 0.208 (sec). Leaf size: 144
ode=3*z[x]^2*D[z[x],x]-a*z[x]^3==x+1;
ic={};
DSolve[{ode,ic},z[x],x,IncludeSingularSolutions->True]
\begin{align*} z(x)&\to e^{\frac {a x}{3}} \sqrt [3]{3 \int _1^x\frac {1}{3} e^{-a K[1]} (K[1]+1)dK[1]+c_1}\\ z(x)&\to -\sqrt [3]{-1} e^{\frac {a x}{3}} \sqrt [3]{3 \int _1^x\frac {1}{3} e^{-a K[1]} (K[1]+1)dK[1]+c_1}\\ z(x)&\to (-1)^{2/3} e^{\frac {a x}{3}} \sqrt [3]{3 \int _1^x\frac {1}{3} e^{-a K[1]} (K[1]+1)dK[1]+c_1} \end{align*}
✓ Sympy. Time used: 15.610 (sec). Leaf size: 196
from sympy import *
x = symbols("x")
a = symbols("a")
z = Function("z")
ode = Eq(-a*z(x)**3 - x + 3*z(x)**2*Derivative(z(x), x) - 1,0)
ics = {}
dsolve(ode,func=z(x),ics=ics)
\[
\left [ z{\left (x \right )} = \sqrt [3]{\left (C_{1} + 3 \left (\begin {cases} - \frac {x e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{6} + \frac {x}{3} & \text {otherwise} \end {cases}\right )\right ) e^{a x}}, \ z{\left (x \right )} = \frac {\sqrt [3]{\left (C_{1} + 3 \left (\begin {cases} - \frac {x e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{6} + \frac {x}{3} & \text {otherwise} \end {cases}\right )\right ) e^{a x}} \left (-1 - \sqrt {3} i\right )}{2}, \ z{\left (x \right )} = \frac {\sqrt [3]{\left (C_{1} + 3 \left (\begin {cases} - \frac {x e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{6} + \frac {x}{3} & \text {otherwise} \end {cases}\right )\right ) e^{a x}} \left (-1 + \sqrt {3} i\right )}{2}\right ]
\]