68.6.17 problem 18

Internal problem ID [17327]
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 : 18
Date solved : Thursday, October 02, 2025 at 02:02:42 PM
CAS classification : [[_homogeneous, `class G`], _exact, _rational]

\begin{align*} 2 t y^{3}+\left (1+3 y^{2} t^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 253

ode:=2*t*y(t)^3+(1+3*t^2*y(t)^2)*diff(y(t),t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= -\frac {12^{{1}/{3}} \left (c_1^{2} 12^{{1}/{3}}-{\left (\left (\sqrt {12 c_1^{2}+81 t^{2}}+9 t \right ) c_1^{2}\right )}^{{2}/{3}}\right )}{6 t c_1 {\left (\left (\sqrt {12 c_1^{2}+81 t^{2}}+9 t \right ) c_1^{2}\right )}^{{1}/{3}}} \\ y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {12 c_1^{2}+81 t^{2}}+9 t \right ) c_1^{2}\right )}^{{2}/{3}}+c_1^{2} 2^{{2}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right )}{12 {\left (\left (\sqrt {12 c_1^{2}+81 t^{2}}+9 t \right ) c_1^{2}\right )}^{{1}/{3}} t c_1} \\ y &= \frac {\left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {12 c_1^{2}+81 t^{2}}+9 t \right ) c_1^{2}\right )}^{{2}/{3}}+c_1^{2} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) 2^{{2}/{3}}\right ) 2^{{2}/{3}} 3^{{1}/{3}}}{12 {\left (\left (\sqrt {12 c_1^{2}+81 t^{2}}+9 t \right ) c_1^{2}\right )}^{{1}/{3}} t c_1} \\ \end{align*}

✓ Mathematica. Time used: 42.613 (sec). Leaf size: 364

ode=2*t*y[t]^3+(1+3*t^2*y[t]^2)*D[y[t],t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {-2 \sqrt [3]{3}+\frac {\sqrt [3]{2} \left (9 c_1 t^4+\sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}\right ){}^{2/3}}{t^2}}{6^{2/3} \sqrt [3]{9 c_1 t^4+\sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}}}\\ y(t)&\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) t^2+i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (18 c_1 t^4+2 \sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}\right ){}^{2/3}}{12 t^2 \sqrt [3]{9 c_1 t^4+\sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}}}\\ y(t)&\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) t^2+\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (18 c_1 t^4+2 \sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}\right ){}^{2/3}}{12 t^2 \sqrt [3]{9 c_1 t^4+\sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}}}\\ y(t)&\to 0 \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*t*y(t)**3 + (3*t**2*y(t)**2 + 1)*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

Timed Out