73.1.13 problem 3 (i)

Internal problem ID [19786]
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, October 02, 2025 at 04:43:29 PM
CAS classification : [_separable]

\begin{align*} 3 x t^{2}-x t +\left (3 t^{3} x^{2}+t^{3} x^{4}\right ) x^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.010 (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: 9.261 (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: 6.959 (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)
 

\[ \left [ x{\left (t \right )} = - \sqrt {-3 - \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = \sqrt {-3 - \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = - \sqrt {-3 + \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = \sqrt {-3 + \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = 0\right ] \]