68.6.16 problem 17
Internal
problem
ID
[17326]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
2.
First
Order
Equations.
Exercises
2.4,
page
57
Problem
number
:
17
Date
solved
:
Thursday, October 02, 2025 at 02:02:38 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 2 y t +\left (t^{2}+y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 209
ode:=2*t*y(t)+(t^2+y(t)^2)*diff(y(t),t) = 0;
dsolve(ode,y(t), singsol=all);
\begin{align*}
y &= -\frac {2 \left (t^{2} c_1 -\frac {\left (4+4 \sqrt {4 t^{6} c_1^{3}+1}\right )^{{2}/{3}}}{4}\right )}{\sqrt {c_1}\, \left (4+4 \sqrt {4 t^{6} c_1^{3}+1}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 t^{6} c_1^{3}+1}\right )^{{1}/{3}}}{4 \sqrt {c_1}}-\frac {\sqrt {c_1}\, \left (i \sqrt {3}-1\right ) t^{2}}{\left (4+4 \sqrt {4 t^{6} c_1^{3}+1}\right )^{{1}/{3}}} \\
y &= \frac {4 i \sqrt {3}\, c_1 \,t^{2}+i \left (4+4 \sqrt {4 t^{6} c_1^{3}+1}\right )^{{2}/{3}} \sqrt {3}+4 t^{2} c_1 -\left (4+4 \sqrt {4 t^{6} c_1^{3}+1}\right )^{{2}/{3}}}{4 \left (4+4 \sqrt {4 t^{6} c_1^{3}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
\end{align*}
✓ Mathematica. Time used: 12.984 (sec). Leaf size: 406
ode=2*t*y[t]+(t^2+y[t]^2)*D[y[t],t]==0;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \frac {\sqrt [3]{\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} t^2}{\sqrt [3]{\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}}}\\ y(t)&\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) t^2}{4 \sqrt [3]{\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}}}\\ y(t)&\to \frac {\left (1-i \sqrt {3}\right ) t^2}{2^{2/3} \sqrt [3]{\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}\\ y(t)&\to 0\\ y(t)&\to \frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{t^6}+\left (1-i \sqrt {3}\right ) t^2}{2 \sqrt [6]{t^6}}\\ y(t)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{t^6}+\left (1+i \sqrt {3}\right ) t^2}{2 \sqrt [6]{t^6}}\\ y(t)&\to \sqrt [6]{t^6}-\frac {\left (t^6\right )^{5/6}}{t^4} \end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(2*t*y(t) + (t**2 + y(t)**2)*Derivative(y(t), t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
Timed Out