74.7.19 problem 19

Internal problem ID [16096]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 19
Date solved : Tuesday, January 28, 2025 at 08:39:46 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

Time used: 2.319 (sec). Leaf size: 235

dsolve(( t*y(t)-y(t)^2 )+( t*(t-3*y(t)) )*diff(y(t),t)=0,y(t), singsol=all)
 

\begin{align*} y &= \frac {{\left (\left (\sqrt {3}\, \sqrt {c_{1}^{4} t^{4}+27}+9\right ) t c_{1} \right )}^{{1}/{3}} 3^{{2}/{3}}}{c_{1} \left (-c_{1}^{2} t^{2} 3^{{1}/{3}}+{\left (\left (\sqrt {3}\, \sqrt {c_{1}^{4} t^{4}+27}+9\right ) t c_{1} \right )}^{{2}/{3}}\right )} \\ y &= -\frac {2 {\left (\left (\sqrt {3}\, \sqrt {c_{1}^{4} t^{4}+27}+9\right ) t c_{1} \right )}^{{1}/{3}} 3^{{2}/{3}}}{c_{1} \left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {c_{1}^{4} t^{4}+27}+9\right ) t c_{1} \right )}^{{2}/{3}}+t^{2} c_{1}^{2} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right )} \\ y &= \frac {2 {\left (\left (\sqrt {3}\, \sqrt {c_{1}^{4} t^{4}+27}+9\right ) t c_{1} \right )}^{{1}/{3}} 3^{{2}/{3}}}{\left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {c_{1}^{4} t^{4}+27}+9\right ) t c_{1} \right )}^{{2}/{3}}+t^{2} \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) c_{1}^{2}\right ) c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.117 (sec). Leaf size: 42

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

\[ \text {Solve}\left [\int _1^{\frac {y(t)}{t}}\frac {3 K[1]-1}{K[1] (2 K[1]-1)}dK[1]=-2 \log (t)+c_1,y(t)\right ] \]