72.1.12 problem 15

Internal problem ID [14533]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.2. page 33
Problem number : 15
Date solved : Thursday, March 13, 2025 at 03:32:45 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

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

\begin{align*} y &= -\frac {1}{2}-\frac {\sqrt {1+4 c_{1} +4 t}}{2} \\ y &= -\frac {1}{2}+\frac {\sqrt {1+4 c_{1} +4 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 49

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

\begin{align*} y(t)\to \frac {1}{2} \left (-1-\sqrt {4 t+1+4 c_1}\right ) \\ y(t)\to \frac {1}{2} \left (-1+\sqrt {4 t+1+4 c_1}\right ) \\ \end{align*}

✓ Sympy. Time used: 0.276 (sec). Leaf size: 32

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

\[ \left [ y{\left (t \right )} = - \frac {\sqrt {C_{1} + 4 t}}{2} - \frac {1}{2}, \ y{\left (t \right )} = \frac {\sqrt {C_{1} + 4 t}}{2} - \frac {1}{2}\right ] \]