90.6.9 problem 9
Internal
problem
ID
[25153]
Book
:
Ordinary
Differential
Equations.
By
William
Adkins
and
Mark
G
Davidson.
Springer.
NY.
2010.
ISBN
978-1-4614-3617-1
Section
:
Chapter
1.
First
order
differential
equations.
Exercises
at
page
83
Problem
number
:
9
Date
solved
:
Thursday, October 02, 2025 at 11:55:09 PM
CAS
classification
:
[[_homogeneous, `class G`], _exact, _rational]
\begin{align*} \left (y^{3}-t \right ) y^{\prime }&=y \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 18
ode:=(y(t)^3-t)*diff(y(t),t) = y(t);
dsolve(ode,y(t), singsol=all);
\[
-\frac {c_1}{y}+t -\frac {y^{3}}{4} = 0
\]
✓ Mathematica. Time used: 46.713 (sec). Leaf size: 996
ode=(y[t]^3-t)*D[y[t],t]== y[t];
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to -\frac {\sqrt {\frac {\left (9 t^2-\sqrt {81 t^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}}}{\sqrt {2} \sqrt [3]{3}}-\frac {1}{2} \sqrt {-\frac {4 \sqrt {2} \sqrt [3]{3} t}{\sqrt {\frac {\left (9 t^2-\sqrt {81 t^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}}}-\frac {2 \sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}{3^{2/3}}-\frac {8 c_1}{\sqrt [3]{27 t^2-3 \sqrt {81 t^4-192 c_1{}^3}}}}\\ y(t)&\to \frac {1}{2} \left (\sqrt {-\frac {4 \sqrt {2} \sqrt [3]{3} t}{\sqrt {\frac {\left (9 t^2-\sqrt {81 t^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}}}-\frac {2 \sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}{3^{2/3}}-\frac {8 c_1}{\sqrt [3]{27 t^2-3 \sqrt {81 t^4-192 c_1{}^3}}}}-\frac {\sqrt {2} \sqrt {\frac {\left (9 t^2-\sqrt {81 t^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}}}{\sqrt [3]{3}}\right )\\ y(t)&\to \frac {\sqrt {\frac {\left (9 t^2-\sqrt {81 t^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}}}{\sqrt {2} \sqrt [3]{3}}-\frac {1}{2} \sqrt {\frac {4 \sqrt {2} \sqrt [3]{3} t}{\sqrt {\frac {\left (9 t^2-\sqrt {81 t^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}}}-\frac {2 \sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}{3^{2/3}}-\frac {8 c_1}{\sqrt [3]{27 t^2-3 \sqrt {81 t^4-192 c_1{}^3}}}}\\ y(t)&\to \frac {1}{2} \left (\frac {\sqrt {2} \sqrt {\frac {\left (9 t^2-\sqrt {81 t^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}}}{\sqrt [3]{3}}+\sqrt {\frac {4 \sqrt {2} \sqrt [3]{3} t}{\sqrt {\frac {\left (9 t^2-\sqrt {81 t^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}}}-\frac {2 \sqrt [3]{9 t^2-\sqrt {81 t^4-192 c_1{}^3}}}{3^{2/3}}-\frac {8 c_1}{\sqrt [3]{27 t^2-3 \sqrt {81 t^4-192 c_1{}^3}}}}\right )\\ y(t)&\to 0 \end{align*}
✓ Sympy. Time used: 36.550 (sec). Leaf size: 648
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq((-t + y(t)**3)*Derivative(y(t), t) - y(t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
\left [ y{\left (t \right )} = - \frac {\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} + 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}}{2} - \frac {\sqrt {\frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} - \frac {8 t}{\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} + 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}} - 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}}{2}, \ y{\left (t \right )} = - \frac {\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} + 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}}{2} + \frac {\sqrt {\frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} - \frac {8 t}{\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} + 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}} - 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}}{2}, \ y{\left (t \right )} = \frac {\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} + 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}}{2} - \frac {\sqrt {\frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} + \frac {8 t}{\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} + 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}} - 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}}{2}, \ y{\left (t \right )} = \frac {\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} + 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}}{2} + \frac {\sqrt {\frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} + \frac {8 t}{\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}} + 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}} - 2 \sqrt [3]{t^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + t^{4}}}}}{2}\right ]
\]