Internal problem ID [11441]
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).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact]
\[ \boxed {\frac {x}{t}+\left (x^{2}+\ln \left (t \right )\right ) x^{\prime }=-t^{3}} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 415
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 t^{8}+72 t^{4} c_{1} +144 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 \ln \left (t \right )}{\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 t^{8}+72 t^{4} c_{1} +144 c_{1}^{2}}\right )^{\frac {1}{3}}} x \left (t \right ) = -\frac {\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 t^{8}+72 t^{4} c_{1} +144 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {\ln \left (t \right )}{\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 t^{8}+72 t^{4} c_{1} +144 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 t^{8}+72 t^{4} c_{1} +144 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 \ln \left (t \right )}{\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 t^{8}+72 t^{4} c_{1} +144 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} x \left (t \right ) = -\frac {\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 t^{8}+72 t^{4} c_{1} +144 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {\ln \left (t \right )}{\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 t^{8}+72 t^{4} c_{1} +144 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 t^{8}+72 t^{4} c_{1} +144 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 \ln \left (t \right )}{\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 t^{8}+72 t^{4} c_{1} +144 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 2.922 (sec). Leaf size: 307
DSolve[t^3+x[t]/t+(x[t]^2+Log[t])*x'[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*}