44.5.72 problem 61

Internal problem ID [7134]
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 : 61
Date solved : Sunday, March 30, 2025 at 11:48:21 AM
CAS classification : [_quadrature]

\begin{align*} m^{\prime }&=-\frac {k}{m^{2}} \end{align*}

With initial conditions

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

✗ Maple

ode:=diff(m(t),t) = -k/m(t)^2; 
ic:=m(0) = m__0; 
dsolve([ode,ic],m(t), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.191 (sec). Leaf size: 62

ode=D[m[t],t]== -k/m[t]^2; 
ic={m[0]==m0}; 
DSolve[{ode,ic},m[t],t,IncludeSingularSolutions->True]
 

\begin{align*} m(t)\to \sqrt [3]{\text {m0}^3-3 k t} \\ m(t)\to -\sqrt [3]{-1} \sqrt [3]{\text {m0}^3-3 k t} \\ m(t)\to (-1)^{2/3} \sqrt [3]{\text {m0}^3-3 k t} \\ \end{align*}

✓ Sympy. Time used: 1.054 (sec). Leaf size: 70

from sympy import * 
t = symbols("t") 
k = symbols("k") 
m = Function("m") 
ode = Eq(k/m(t)**2 + Derivative(m(t), t),0) 
ics = {m(0): m__0} 
dsolve(ode,func=m(t),ics=ics)
 

\[ \left [ m{\left (t \right )} = \sqrt [3]{- 3 k t + \left (m^{0}\right )^{3}}, \ m{\left (t \right )} = \frac {\left (- \sqrt [3]{3} - 3^{\frac {5}{6}} i\right ) \sqrt [3]{- k t + \frac {\left (m^{0}\right )^{3}}{3}}}{2}, \ m{\left (t \right )} = \frac {\left (- \sqrt [3]{3} + 3^{\frac {5}{6}} i\right ) \sqrt [3]{- k t + \frac {\left (m^{0}\right )^{3}}{3}}}{2}\right ] \]