85.1.4 problem 1 (e)

Internal problem ID [22409]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 1. Differential equations in general. A Exercises at page 12
Problem number : 1 (e)
Date solved : Thursday, October 02, 2025 at 08:38:25 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} r^{\prime }&=\sqrt {r t} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 66

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

\[ \frac {3 \left (c_1 \,t^{3}-9 r c_1 +1\right ) \sqrt {r t}-t^{2} \left (c_1 \,t^{3}-9 r c_1 -1\right )}{\left (t^{3}-9 r\right ) \left (t^{2}-3 \sqrt {r t}\right )} = 0 \]

✓ Mathematica. Time used: 0.094 (sec). Leaf size: 28

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

\begin{align*} r(t)&\to \frac {1}{36} \left (2 t^{3/2}+3 c_1\right ){}^2\\ r(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.301 (sec). Leaf size: 22

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

\[ r{\left (t \right )} = \frac {C_{1}^{2}}{4} - \frac {C_{1} \sqrt {t^{3}}}{3} + \frac {t^{3}}{9} \]