63.5.33 problem 16-b(ii)

Internal problem ID [13107]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.4.1. Integrating factors. Exercises page 41
Problem number : 16-b(ii)
Date solved : Tuesday, January 28, 2025 at 04:52:37 AM
CAS classification : [_exact]

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

✓ Solution by Maple

Time used: 0.065 (sec). Leaf size: 305

dsolve(t^3+x(t)/t+(x(t)^2+ln(t))*diff(x(t),t)=0,x(t), singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{{2}/{3}}-4 \ln \left (t \right )}{2 \left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{{1}/{3}}} \\ x \left (t \right ) &= \frac {i \left (-\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{{2}/{3}}-4 \ln \left (t \right )\right ) \sqrt {3}-\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (t \right )}{4 \left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{{1}/{3}}} \\ x \left (t \right ) &= \frac {i \left (\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (t \right )\right ) \sqrt {3}-\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (t \right )}{4 \left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.913 (sec). Leaf size: 307

DSolve[t^3+x[t]/t+(x[t]^2+Log[t])*D[x[t],t]==0,x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {-4 \log (t)+\left (-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1\right ){}^{2/3}}{2 \sqrt [3]{-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1}} \\ x(t)\to \frac {i \left (\sqrt {3}+i\right ) \left (-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1\right ){}^{2/3}+\left (4+4 i \sqrt {3}\right ) \log (t)}{4 \sqrt [3]{-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1}} \\ x(t)\to \frac {\left (-1-i \sqrt {3}\right ) \left (-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1\right ){}^{2/3}+\left (4-4 i \sqrt {3}\right ) \log (t)}{4 \sqrt [3]{-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1}} \\ \end{align*}