76.5.19 problem 19

Internal problem ID [17368]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 19
Date solved : Monday, March 31, 2025 at 04:09:19 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y+\sqrt {y} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 16

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

\[ 1-{\mathrm e}^{\frac {t}{2}} c_1 +\sqrt {y} = 0 \]

✓ Mathematica. Time used: 0.189 (sec). Leaf size: 29

ode=D[y[t],t]==y[t]+Sqrt[ y[t] ]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to \left (-1+e^{\frac {t+c_1}{2}}\right ){}^2 \\ y(t)\to 0 \\ y(t)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.411 (sec). Leaf size: 19

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

\[ y{\left (t \right )} = e^{- C_{1} + t} - 2 e^{- \frac {C_{1}}{2} + \frac {t}{2}} + 1 \]