73.1.23 problem 4 (v)

Internal problem ID [19796]
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 : 4 (v)
Date solved : Thursday, October 02, 2025 at 04:43:49 PM
CAS classification : [_separable]

\begin{align*} x^{\prime }+2 x t +t x^{4}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 121

ode:=diff(x(t),t)+2*t*x(t)+t*x(t)^4 = 0; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x &= \frac {2^{{1}/{3}} {\left (\left (2 \,{\mathrm e}^{3 t^{2}} c_1 -1\right )^{2}\right )}^{{1}/{3}}}{2 \,{\mathrm e}^{3 t^{2}} c_1 -1} \\ x &= -\frac {2^{{1}/{3}} {\left (\left (2 \,{\mathrm e}^{3 t^{2}} c_1 -1\right )^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4 \,{\mathrm e}^{3 t^{2}} c_1 -2} \\ x &= \frac {2^{{1}/{3}} {\left (\left (2 \,{\mathrm e}^{3 t^{2}} c_1 -1\right )^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 \,{\mathrm e}^{3 t^{2}} c_1 -2} \\ \end{align*}

✓ Mathematica. Time used: 10.652 (sec). Leaf size: 177

ode=D[x[t],t]+2*t*x[t]+t*x[t]^4==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {\sqrt [3]{-2} e^{2 c_1}}{\sqrt [3]{e^{3 t^2}-e^{6 c_1}}}\\ x(t)&\to \frac {\sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{e^{3 t^2}-e^{6 c_1}}}\\ x(t)&\to \frac {(-1)^{2/3} \sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{e^{3 t^2}-e^{6 c_1}}}\\ x(t)&\to 0\\ x(t)&\to \sqrt [3]{-2}\\ x(t)&\to -\sqrt [3]{2}\\ x(t)&\to -(-1)^{2/3} \sqrt [3]{2}\\ x(t)&\to \frac {1-i \sqrt {3}}{2^{2/3}} \end{align*}

✓ Sympy. Time used: 2.032 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t*x(t)**4 + 2*t*x(t) + Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ \left [ x{\left (t \right )} = \sqrt [3]{2} \sqrt [3]{- \frac {C_{1}}{C_{1} - e^{3 t^{2}}}}, \ x{\left (t \right )} = \frac {\sqrt [3]{2} \sqrt [3]{- \frac {C_{1}}{C_{1} - e^{3 t^{2}}}} \left (-1 - \sqrt {3} i\right )}{2}, \ x{\left (t \right )} = \frac {\sqrt [3]{2} \sqrt [3]{- \frac {C_{1}}{C_{1} - e^{3 t^{2}}}} \left (-1 + \sqrt {3} i\right )}{2}\right ] \]