Internal problem ID [14305]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number: 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\[ \boxed {2 t y+\left (y^{2}+t^{2}\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 209
dsolve(2*t*y(t)+(t^2+y(t)^2)*diff(y(t),t)=0,y(t), singsol=all)
\begin{align*} y \left (t \right ) &= -\frac {2 \left (c_{1} t^{2}-\frac {\left (4+4 \sqrt {4 c_{1}^{3} t^{6}+1}\right )^{\frac {2}{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4+4 \sqrt {4 c_{1}^{3} t^{6}+1}\right )^{\frac {1}{3}}} \\ y \left (t \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 c_{1}^{3} t^{6}+1}\right )^{\frac {1}{3}}}{4 \sqrt {c_{1}}}-\frac {\sqrt {c_{1}}\, t^{2} \left (i \sqrt {3}-1\right )}{\left (4+4 \sqrt {4 c_{1}^{3} t^{6}+1}\right )^{\frac {1}{3}}} \\ y \left (t \right ) &= \frac {4 i \sqrt {3}\, c_{1} t^{2}+i \sqrt {3}\, \left (4+4 \sqrt {4 c_{1}^{3} t^{6}+1}\right )^{\frac {2}{3}}+4 c_{1} t^{2}-\left (4+4 \sqrt {4 c_{1}^{3} t^{6}+1}\right )^{\frac {2}{3}}}{4 \left (4+4 \sqrt {4 c_{1}^{3} t^{6}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.781 (sec). Leaf size: 406
DSolve[2*t*y[t]+(t^2+y[t]^2)*y'[t]==0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {\sqrt [3]{\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} t^2}{\sqrt [3]{\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(t)\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) t^2}{4 \sqrt [3]{\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(t)\to \frac {\left (1-i \sqrt {3}\right ) t^2}{2^{2/3} \sqrt [3]{\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 t^6+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\ y(t)\to 0 \\ y(t)\to \frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{t^6}+\left (1-i \sqrt {3}\right ) t^2}{2 \sqrt [6]{t^6}} \\ y(t)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{t^6}+\left (1+i \sqrt {3}\right ) t^2}{2 \sqrt [6]{t^6}} \\ y(t)\to \sqrt [6]{t^6}-\frac {\left (t^6\right )^{5/6}}{t^4} \\ \end{align*}