79.1.23 problem 4 (v)

Internal problem ID [18510]
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 : Tuesday, January 28, 2025 at 11:51:30 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 121

dsolve(diff(x(t),t)+2*t*x(t)+t*x(t)^4=0,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 {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} {\left (\left (2 \,{\mathrm e}^{3 t^{2}} c_{1} -1\right )^{2}\right )}^{{1}/{3}}}{4 \,{\mathrm e}^{3 t^{2}} c_{1} -2} \\ x &= \frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} {\left (\left (2 \,{\mathrm e}^{3 t^{2}} c_{1} -1\right )^{2}\right )}^{{1}/{3}}}{4 \,{\mathrm e}^{3 t^{2}} c_{1} -2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 10.919 (sec). Leaf size: 177

DSolve[D[x[t],t]+2*t*x[t]+t*x[t]^4==0,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*}