Internal problem ID [7121]
Internal file name [OUTPUT/6107_Sunday_June_05_2022_04_22_06_PM_97271531/index.tex]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 77.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {x^{\prime }-4 A k \left (\frac {x}{A}\right )^{\frac {3}{4}}+3 k x=0} \]
Integrating both sides gives \begin{align*} \int \frac {1}{4 A k \left (\frac {x}{A}\right )^{\frac {3}{4}}-3 k x}d x &= \int d t \\ -\frac {4 \ln \left (3 \left (\frac {x}{A}\right )^{\frac {1}{4}}-4\right )}{3 k}&=t +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} -\frac {4 \ln \left (3 \left (\frac {x}{A}\right )^{\frac {1}{4}}-4\right )}{3 k} &= t +c_{1} \\ \end{align*}
Verification of solutions
\[ -\frac {4 \ln \left (3 \left (\frac {x}{A}\right )^{\frac {1}{4}}-4\right )}{3 k} = t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }-4 A k \left (\frac {x}{A}\right )^{\frac {3}{4}}+3 k x=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=4 A k \left (\frac {x}{A}\right )^{\frac {3}{4}}-3 k x \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{4 A k \left (\frac {x}{A}\right )^{\frac {3}{4}}-3 k x}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{4 A k \left (\frac {x}{A}\right )^{\frac {3}{4}}-3 k x}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {-\frac {\ln \left (256 A -81 x\right )}{3}+\frac {\ln \left (9 \sqrt {\frac {x}{A}}+16\right )}{3}-\frac {\ln \left (9 \sqrt {\frac {x}{A}}-16\right )}{3}+\frac {2 \ln \left (3 \left (\frac {x}{A}\right )^{\frac {1}{4}}+4\right )}{3}-\frac {2 \ln \left (3 \left (\frac {x}{A}\right )^{\frac {1}{4}}-4\right )}{3}}{k}=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & \left \{x=\frac {-\frac {16 \left (-A^{3} {\mathrm e}^{c_{1} k} {\mathrm e}^{k t}\right )^{\frac {3}{4}} {\mathrm e}^{3 \left (t +c_{1} \right ) k}}{A^{2} \left ({\mathrm e}^{c_{1} k}\right )^{3} \left ({\mathrm e}^{k t}\right )^{3}}+\frac {96 \,{\mathrm e}^{3 \left (t +c_{1} \right ) k} \sqrt {-A^{3} {\mathrm e}^{c_{1} k} {\mathrm e}^{k t}}}{A \left ({\mathrm e}^{c_{1} k}\right )^{2} \left ({\mathrm e}^{k t}\right )^{2}}-\frac {256 \,{\mathrm e}^{3 \left (t +c_{1} \right ) k} \left (-A^{3} {\mathrm e}^{c_{1} k} {\mathrm e}^{k t}\right )^{\frac {1}{4}}}{{\mathrm e}^{c_{1} k} {\mathrm e}^{k t}}+256 A \,{\mathrm e}^{3 \left (t +c_{1} \right ) k}-1}{81 \,{\mathrm e}^{3 \left (t +c_{1} \right ) k}}, x=\frac {\frac {16 \left (-A^{3} {\mathrm e}^{c_{1} k} {\mathrm e}^{k t}\right )^{\frac {3}{4}} {\mathrm e}^{3 \left (t +c_{1} \right ) k}}{A^{2} \left ({\mathrm e}^{c_{1} k}\right )^{3} \left ({\mathrm e}^{k t}\right )^{3}}+\frac {96 \,{\mathrm e}^{3 \left (t +c_{1} \right ) k} \sqrt {-A^{3} {\mathrm e}^{c_{1} k} {\mathrm e}^{k t}}}{A \left ({\mathrm e}^{c_{1} k}\right )^{2} \left ({\mathrm e}^{k t}\right )^{2}}+\frac {256 \,{\mathrm e}^{3 \left (t +c_{1} \right ) k} \left (-A^{3} {\mathrm e}^{c_{1} k} {\mathrm e}^{k t}\right )^{\frac {1}{4}}}{{\mathrm e}^{c_{1} k} {\mathrm e}^{k t}}+256 A \,{\mathrm e}^{3 \left (t +c_{1} \right ) k}-1}{81 \,{\mathrm e}^{3 \left (t +c_{1} \right ) k}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (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)
\[ \frac {\ln \left (9 \sqrt {\frac {x \left (t \right )}{A}}-16\right )-\ln \left (9 \sqrt {\frac {x \left (t \right )}{A}}+16\right )+2 \ln \left (3 \left (\frac {x \left (t \right )}{A}\right )^{\frac {1}{4}}-4\right )-2 \ln \left (3 \left (\frac {x \left (t \right )}{A}\right )^{\frac {1}{4}}+4\right )+\ln \left (256 A -81 x \left (t \right )\right )+\left (3 t +3 c_{1} \right ) k}{3 k} = 0 \]
✓ Solution by Mathematica
Time used: 0.409 (sec). Leaf size: 51
DSolve[x'[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*}