57.4.15 problem 5

Internal problem ID [14339]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.3.1 Separable equations. Exercises page 26
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:31:51 AM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 15

ode:=diff(y(t),t) = 1/(2*y(t)+1); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {1}{2}+\frac {\sqrt {9+4 t}}{2} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 20

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

\begin{align*} y(t)&\to \frac {1}{2} \left (\sqrt {4 t+9}-1\right ) \end{align*}

✓ Sympy. Time used: 0.206 (sec). Leaf size: 15

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

\[ y{\left (t \right )} = \frac {\sqrt {4 t + 9}}{2} - \frac {1}{2} \]