74.6.16 problem 17

Internal problem ID [16039]
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
Date solved : Tuesday, January 28, 2025 at 08:28:39 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.158 (sec). Leaf size: 205

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

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

✓ Solution by Mathematica

Time used: 0.144 (sec). Leaf size: 42

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

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