79.1.13 problem 3 (i)
Internal
problem
ID
[18421]
Book
:
Elementary
Differential
Equations.
By
R.L.E.
Schwarzenberger.
Chapman
and
Hall.
London.
First
Edition
(1969)
Section
:
Chapter
3.
Solutions
of
first-order
equations.
Exercises
at
page
47
Problem
number
:
3
(i)
Date
solved
:
Thursday, March 13, 2025 at 11:56:12 AM
CAS
classification
:
[_separable]
\begin{align*} 3 t^{2} x-t x+\left (3 t^{3} x^{2}+t^{3} x^{4}\right ) x^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 139
ode:=3*t^2*x(t)-t*x(t)+(3*t^3*x(t)^2+t^3*x(t)^4)*diff(x(t),t) = 0;
dsolve(ode,x(t), singsol=all);
\begin{align*}
x &= 0 \\
x &= \frac {\sqrt {-3 t^{2}+2 t \sqrt {-t \left (1+3 \ln \left (t \right ) t +c_{1} t \right )}}}{t} \\
x &= \frac {\sqrt {-3 t^{2}-2 t \sqrt {-t \left (1+3 \ln \left (t \right ) t +c_{1} t \right )}}}{t} \\
x &= -\frac {\sqrt {-3 t^{2}+2 t \sqrt {-t \left (1+3 \ln \left (t \right ) t +c_{1} t \right )}}}{t} \\
x &= -\frac {\sqrt {-3 t^{2}-2 t \sqrt {-t \left (1+3 \ln \left (t \right ) t +c_{1} t \right )}}}{t} \\
\end{align*}
✓ Mathematica. Time used: 6.903 (sec). Leaf size: 157
ode=(3*t^2*x[t]-t*x[t])+(3*t^3*x[t]^2+t^3*x[t]^4)*D[x[t],t]==0;
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to 0 \\
x(t)\to -\sqrt {-3-\frac {\sqrt {9 t-12 t \log (t)+4 c_1 t-4}}{\sqrt {t}}} \\
x(t)\to \sqrt {-3-\frac {\sqrt {9 t-12 t \log (t)+4 c_1 t-4}}{\sqrt {t}}} \\
x(t)\to -\sqrt {-3+\frac {\sqrt {9 t-12 t \log (t)+4 c_1 t-4}}{\sqrt {t}}} \\
x(t)\to \sqrt {-3+\frac {\sqrt {9 t-12 t \log (t)+4 c_1 t-4}}{\sqrt {t}}} \\
x(t)\to 0 \\
\end{align*}
✓ Sympy. Time used: 9.627 (sec). Leaf size: 126
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(3*t**2*x(t) - t*x(t) + (t**3*x(t)**4 + 3*t**3*x(t)**2)*Derivative(x(t), t),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
\[
\left [ x{\left (t \right )} = - \sqrt {-3 - \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = \sqrt {-3 - \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = - \sqrt {-3 + \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = \sqrt {-3 + \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = 0\right ]
\]