68.7.19 problem 19
Internal
problem
ID
[17383]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
2.
First
Order
Equations.
Exercises
2.5,
page
64
Problem
number
:
19
Date
solved
:
Thursday, October 02, 2025 at 02:15:30 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y t -y^{2}+t \left (t -3 y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.019 (sec). Leaf size: 494
ode:=t*y(t)-y(t)^2+t*(t-3*y(t))*diff(y(t),t) = 0;
dsolve(ode,y(t), singsol=all);
\begin{align*}
y &= \frac {c_1^{4} t^{4}+t^{2} c_1^{2} {\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{1}/{3}}+{\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{2}/{3}}}{6 t \,c_1^{2} {\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, c_1^{4} t^{4}-c_1^{4} t^{4}+2 t^{2} c_1^{2} {\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{1}/{3}}-i {\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{2}/{3}} \sqrt {3}-{\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{2}/{3}}}{12 t \,c_1^{2} {\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{1}/{3}}} \\
y &= -\frac {i \sqrt {3}\, c_1^{4} t^{4}+c_1^{4} t^{4}-2 t^{2} c_1^{2} {\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{1}/{3}}-i {\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{2}/{3}} \sqrt {3}+{\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{2}/{3}}}{12 t \,c_1^{2} {\left (\left (c_1^{4} t^{4}+6 \sqrt {3}\, \sqrt {c_1^{4} t^{4}+27}+54\right ) c_1^{2} t^{2}\right )}^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 60.092 (sec). Leaf size: 373
ode=( t*y[t]-y[t]^2 )+( t*(t-3*y[t]) )*D[y[t],t]==0;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \frac {1}{6} \left (\sqrt [3]{\frac {t^4+6 \sqrt {81 e^{4 c_1}-3 e^{2 c_1} t^4}-54 e^{2 c_1}}{t}}+\frac {t^2}{\sqrt [3]{\frac {t^4+6 \sqrt {81 e^{4 c_1}-3 e^{2 c_1} t^4}-54 e^{2 c_1}}{t}}}+t\right )\\ y(t)&\to \frac {1}{12} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {t^4+6 \sqrt {81 e^{4 c_1}-3 e^{2 c_1} t^4}-54 e^{2 c_1}}{t}}-\frac {i \left (\sqrt {3}-i\right ) t^2}{\sqrt [3]{\frac {t^4+6 \sqrt {81 e^{4 c_1}-3 e^{2 c_1} t^4}-54 e^{2 c_1}}{t}}}+2 t\right )\\ y(t)&\to \frac {1}{12} \left (-\left (1+i \sqrt {3}\right ) \sqrt [3]{\frac {t^4+6 \sqrt {81 e^{4 c_1}-3 e^{2 c_1} t^4}-54 e^{2 c_1}}{t}}+\frac {i \left (\sqrt {3}+i\right ) t^2}{\sqrt [3]{\frac {t^4+6 \sqrt {81 e^{4 c_1}-3 e^{2 c_1} t^4}-54 e^{2 c_1}}{t}}}+2 t\right ) \end{align*}
✓ Sympy. Time used: 33.303 (sec). Leaf size: 352
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(t*(t - 3*y(t))*Derivative(y(t), t) + t*y(t) - y(t)**2,0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
\left [ y{\left (t \right )} = \frac {\frac {2 t^{2}}{\sqrt [3]{\frac {54 C_{1}}{t} - t^{3} + 6 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{t^{2}} - t^{2}\right )}}} + t - \sqrt {3} i t - \sqrt [3]{\frac {54 C_{1}}{t} - t^{3} + 6 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{t^{2}} - t^{2}\right )}} - \sqrt {3} i \sqrt [3]{\frac {54 C_{1}}{t} - t^{3} + 6 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{t^{2}} - t^{2}\right )}}}{6 \left (1 - \sqrt {3} i\right )}, \ y{\left (t \right )} = \frac {\frac {2 t^{2}}{\sqrt [3]{\frac {54 C_{1}}{t} - t^{3} + 6 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{t^{2}} - t^{2}\right )}}} + t + \sqrt {3} i t - \sqrt [3]{\frac {54 C_{1}}{t} - t^{3} + 6 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{t^{2}} - t^{2}\right )}} + \sqrt {3} i \sqrt [3]{\frac {54 C_{1}}{t} - t^{3} + 6 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{t^{2}} - t^{2}\right )}}}{6 \left (1 + \sqrt {3} i\right )}, \ y{\left (t \right )} = - \frac {t^{2}}{6 \sqrt [3]{\frac {54 C_{1}}{t} - t^{3} + 6 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{t^{2}} - t^{2}\right )}}} + \frac {t}{6} - \frac {\sqrt [3]{\frac {54 C_{1}}{t} - t^{3} + 6 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{t^{2}} - t^{2}\right )}}}{6}\right ]
\]