68.4.2 problem 2

Internal problem ID [17182]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 2
Date solved : Thursday, October 02, 2025 at 01:49:39 PM
CAS classification : [_separable]

\begin{align*} \frac {1}{2 \sqrt {t}}+y^{2} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 57

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

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

✓ Mathematica. Time used: 1.735 (sec). Leaf size: 79

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

\begin{align*} y(t)&\to -\sqrt [3]{-3} \sqrt [3]{-\sqrt {t}+c_1}\\ y(t)&\to \sqrt [3]{3} \sqrt [3]{-\sqrt {t}+c_1}\\ y(t)&\to (-1)^{2/3} \sqrt [3]{3} \sqrt [3]{-\sqrt {t}+c_1} \end{align*}

✓ Sympy. Time used: 0.852 (sec). Leaf size: 66

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

\[ \left [ y{\left (t \right )} = \sqrt [3]{C_{1} - 3 \sqrt {t}}, \ y{\left (t \right )} = \frac {\left (- \sqrt [3]{3} - 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} - \sqrt {t}}}{2}, \ y{\left (t \right )} = \frac {\left (- \sqrt [3]{3} + 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} - \sqrt {t}}}{2}\right ] \]