23.1.634 problem 628
Internal
problem
ID
[5241]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
628
Date
solved
:
Tuesday, September 30, 2025 at 11:59:49 AM
CAS
classification
:
[_rational, _Bernoulli]
\begin{align*} 3 y^{2} y^{\prime }&=1+x +a y^{3} \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 104
ode:=3*y(x)^2*diff(y(x),x) = 1+x+a*y(x)^3;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {{\left (\left ({\mathrm e}^{a x} c_1 \,a^{2}-1+a \left (-x -1\right )\right ) a \right )}^{{1}/{3}}}{a} \\
y &= -\frac {{\left (\left ({\mathrm e}^{a x} c_1 \,a^{2}-1+a \left (-x -1\right )\right ) a \right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 a} \\
y &= \frac {{\left (\left ({\mathrm e}^{a x} c_1 \,a^{2}-1+a \left (-x -1\right )\right ) a \right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 a} \\
\end{align*}
✓ Mathematica. Time used: 0.246 (sec). Leaf size: 144
ode=3*y[x]^2*D[y[x],x]==1+x+a*y[x]^3;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(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}\\ y(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}\\ y(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: 15.886 (sec). Leaf size: 196
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*y(x)**3 - x + 3*y(x)**2*Derivative(y(x), x) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{\left (C_{1} + 3 \left (\begin {cases} - \frac {x e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{6} + \frac {x}{3} & \text {otherwise} \end {cases}\right )\right ) e^{a x}}, \ y{\left (x \right )} = \frac {\sqrt [3]{\left (C_{1} + 3 \left (\begin {cases} - \frac {x e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{6} + \frac {x}{3} & \text {otherwise} \end {cases}\right )\right ) e^{a x}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{\left (C_{1} + 3 \left (\begin {cases} - \frac {x e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a} - \frac {e^{- a x}}{3 a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{6} + \frac {x}{3} & \text {otherwise} \end {cases}\right )\right ) e^{a x}} \left (-1 + \sqrt {3} i\right )}{2}\right ]
\]