56.1.77 problem 77

Internal problem ID [8789]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 77
Date solved : Monday, January 27, 2025 at 04:58:39 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 85

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

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

✓ Solution by Mathematica

Time used: 0.373 (sec). Leaf size: 51

DSolve[D[x[t],t]==4*A*k*(x[t]/A)^(3/4)-3*k*x[t],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*}