90.3.7 problem 7
Internal
problem
ID
[25071]
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
41
Problem
number
:
7
Date
solved
:
Thursday, October 02, 2025 at 11:48:18 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} \left (t^{2}+3 y^{2}\right ) y^{\prime }&=-2 y t \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 189
ode:=(t^2+3*y(t)^2)*diff(y(t),t) = -2*t*y(t);
dsolve(ode,y(t), singsol=all);
\begin{align*}
y &= \frac {-12 t^{2} c_1 +\left (108+12 \sqrt {12 t^{6} c_1^{3}+81}\right )^{{2}/{3}}}{6 \left (108+12 \sqrt {12 t^{6} c_1^{3}+81}\right )^{{1}/{3}} \sqrt {c_1}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (108+12 \sqrt {12 t^{6} c_1^{3}+81}\right )^{{1}/{3}}}{12 \sqrt {c_1}}-\frac {\sqrt {c_1}\, t^{2} \left (i \sqrt {3}-1\right )}{\left (108+12 \sqrt {12 t^{6} c_1^{3}+81}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {12 t^{6} c_1^{3}+81}\right )^{{1}/{3}}}{12 \sqrt {c_1}}+\frac {\sqrt {c_1}\, \left (1+i \sqrt {3}\right ) t^{2}}{\left (108+12 \sqrt {12 t^{6} c_1^{3}+81}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 28.067 (sec). Leaf size: 442
ode=(t^2+3*y[t]^2)*D[y[t],{t,1}]==-2*t*y[t];
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \frac {-2 \sqrt [3]{3} t^2+\sqrt [3]{2} \left (\sqrt {12 t^6+81 e^{2 c_1}}+9 e^{c_1}\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {12 t^6+81 e^{2 c_1}}+9 e^{c_1}}}\\ y(t)&\to \frac {i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (\sqrt {12 t^6+81 e^{2 c_1}}+9 e^{c_1}\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) t^2}{12 \sqrt [3]{\sqrt {12 t^6+81 e^{2 c_1}}+9 e^{c_1}}}\\ y(t)&\to \frac {2^{2/3} \sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (\sqrt {12 t^6+81 e^{2 c_1}}+9 e^{c_1}\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) t^2}{12 \sqrt [3]{\sqrt {12 t^6+81 e^{2 c_1}}+9 e^{c_1}}}\\ y(t)&\to 0\\ y(t)&\to \frac {\sqrt [3]{t^6}-t^2}{\sqrt {3} \sqrt [6]{t^6}}\\ y(t)&\to \frac {\left (\sqrt {3}-3 i\right ) t^2-\left (\sqrt {3}+3 i\right ) \sqrt [3]{t^6}}{6 \sqrt [6]{t^6}}\\ y(t)&\to \frac {\left (\sqrt {3}+3 i\right ) t^2-\left (\sqrt {3}-3 i\right ) \sqrt [3]{t^6}}{6 \sqrt [6]{t^6}} \end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(2*t*y(t) + (t**2 + 3*y(t)**2)*Derivative(y(t), t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
Timed Out