38.5.74 problem 63

Internal problem ID [8422]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 63
Date solved : Tuesday, September 30, 2025 at 05:36:21 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=k \left (A -x\right )^{2} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 18

ode:=diff(x(t),t) = k*(A-x(t))^2; 
ic:=[x(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {t k \,A^{2}}{t k A +1} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 19

ode=D[x[t],t]== k*(A-x[t])^2; 
ic={x[0]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {A^2 k t}{A k t+1} \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 14

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

\[ x{\left (t \right )} = \frac {A k t}{k t + \frac {1}{A}} \]