76.10.21 problem 22 (iii)

Internal problem ID [17464]
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, March 13, 2025 at 10:08:45 AM
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: 0.443 (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,ic],x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.024 (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]
 

\[ x(t)\to \frac {3}{2} \sec ^2\left (\frac {1}{4} (2 t+\pi )\right ) \]

✓ Sympy. Time used: 0.525 (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)
 

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