50.1.77 problem 77

Internal problem ID [10075]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 77
Date solved : Tuesday, September 30, 2025 at 06:52:03 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=4 A k \left (\frac {x}{A}\right )^{{3}/{4}}-3 k x \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 85

ode:=diff(x(t),t) = 4*A*k*(x(t)/A)^(3/4)-3*k*x(t); 
dsolve(ode,x(t), singsol=all);
 

\[ \frac {\ln \left (9 \sqrt {\frac {x}{A}}-16\right )-\ln \left (9 \sqrt {\frac {x}{A}}+16\right )+2 \ln \left (3 \left (\frac {x}{A}\right )^{{1}/{4}}-4\right )-2 \ln \left (3 \left (\frac {x}{A}\right )^{{1}/{4}}+4\right )+\ln \left (256 A -81 x\right )+\left (3 t +3 c_1 \right ) k}{3 k} = 0 \]

✓ Mathematica. Time used: 0.234 (sec). Leaf size: 51

ode=D[x[t],t]==4*A*k*(x[t]/A)^(3/4)-3*k*x[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{81} A e^{-3 k t} \left (4 e^{\frac {3 k t}{4}}+e^{\frac {3 c_1}{4}}\right ){}^4\\ x(t)&\to 0\\ x(t)&\to \frac {256 A}{81} \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
A = symbols("A") 
k = symbols("k") 
x = Function("x") 
ode = Eq(-4*A*k*(x(t)/A)**(3/4) + 3*k*x(t) + Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

TypeError : < not supported between instances of NoneType and x