70.10.21 problem 22 (iii)

Internal problem ID [18830]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 22 (iii)
Date solved : Thursday, October 02, 2025 at 03:31:13 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=\frac {x \sqrt {6 x-9}}{3} \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.139 (sec). Leaf size: 16

ode:=diff(x(t),t) = 1/3*x(t)*(6*x(t)-9)^(1/2); 
ic:=[x(0) = 3]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {3 \sec \left (\frac {t}{2}+\frac {\pi }{4}\right )^{2}}{2} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 21

ode=D[x[t],t]==1/3*x[t]*Sqrt[6*x[t]-9]; 
ic={x[0]==3}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {3}{2} \sec ^2\left (\frac {1}{4} (2 t+\pi )\right ) \end{align*}

✓ Sympy. Time used: 0.337 (sec). Leaf size: 73

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-sqrt(6*x(t) - 9)*x(t)/3 + Derivative(x(t), t),0) 
ics = {x(0): 3} 
dsolve(ode,func=x(t),ics=ics)
 

\[ \begin {cases} \frac {2 \sqrt {3} i \operatorname {acosh}{\left (\frac {\sqrt {6}}{2 \sqrt {x{\left (t \right )}}} \right )}}{3} & \text {for}\: \frac {1}{\left |{x{\left (t \right )}}\right |} > \frac {2}{3} \\- \frac {2 \sqrt {3} \operatorname {asin}{\left (\frac {\sqrt {6}}{2 \sqrt {x{\left (t \right )}}} \right )}}{3} & \text {otherwise} \end {cases} = \frac {\sqrt {3} t}{3} - \frac {\sqrt {3} \pi }{6} \]