Internal problem ID [11625]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 2, section 2.2 (Separable equations). Exercises page 47
Problem number: 12.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\[ \boxed {\left (2 s^{2}+2 s t +t^{2}\right ) s^{\prime }+s^{2}+2 s t=t^{2}} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 348
dsolve((2*s(t)^2+2*s(t)*t+t^2)*diff(s(t),t)+(s(t)^2+2*s(t)*t-t^2)=0,s(t), singsol=all)
\begin{align*} s \left (t \right ) &= \frac {\left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}}-\frac {t^{2} c_{1}^{2}}{\left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}}}-c_{1} t}{2 c_{1}} \\ s \left (t \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {2}{3}}+c_{1} t \left (2 \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) c_{1} t \right )}{4 \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}} c_{1}} \\ s \left (t \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {2}{3}}+\left (-2 \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}}+c_{1} t \left (1+i \sqrt {3}\right )\right ) c_{1} t}{4 \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 48.03 (sec). Leaf size: 616
DSolve[(2*s[t]^2+2*s[t]*t+t^2)*s'[t]+(s[t]^2+2*s[t]*t-t^2)==0,s[t],t,IncludeSingularSolutions -> True]
\begin{align*} s(t)\to \frac {1}{2} \left (\sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}-\frac {t^2}{\sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}}-t\right ) \\ s(t)\to \frac {1}{8} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}+\frac {2 \left (1+i \sqrt {3}\right ) t^2}{\sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}}-4 t\right ) \\ s(t)\to \frac {1}{8} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}+\frac {2 \left (1-i \sqrt {3}\right ) t^2}{\sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}}-4 t\right ) \\ s(t)\to \frac {1}{2} \left (\sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}-\frac {t^2}{\sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}}-t\right ) \\ s(t)\to \frac {1}{4} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}+\frac {\left (1-i \sqrt {3}\right ) t^2}{\sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}}-2 t\right ) \\ s(t)\to \frac {1}{4} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}+\frac {\left (1+i \sqrt {3}\right ) t^2}{\sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}}-2 t\right ) \\ \end{align*}