Internal problem ID [12012]
Internal file name [OUTPUT/10665_Saturday_September_02_2023_02_57_53_PM_72234192/index.tex]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 10, Two tricks for nonlinear equations. Exercises page 97
Problem number: 10.5.
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]
\[ \boxed {x^{\prime }-k x+x^{2}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{x k -x^{2}}d x &= t +c_{1}\\ \frac {\ln \left (x \right )-\ln \left (-k +x \right )}{k}&=t +c_{1} \end {align*}
Solving for \(x\) gives these solutions \begin {align*} x_1&=\frac {k \,{\mathrm e}^{c_{1} k +t k}}{-1+{\mathrm e}^{c_{1} k +t k}}\\ &=\frac {k c_{1} {\mathrm e}^{t k}}{-1+c_{1} {\mathrm e}^{t k}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= \frac {k c_{1} {\mathrm e}^{t k}}{-1+c_{1} {\mathrm e}^{t k}} \\ \end{align*}
Verification of solutions
\[ x = \frac {k c_{1} {\mathrm e}^{t k}}{-1+c_{1} {\mathrm e}^{t k}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }-k x+x^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=k x-x^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{k x-x^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{k x-x^{2}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (x-k \right )}{k}+\frac {\ln \left (x\right )}{k}=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=\frac {k \,{\mathrm e}^{c_{1} k +t k}}{-1+{\mathrm e}^{c_{1} k +t k}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 18
dsolve(diff(x(t),t)=k*x(t)-x(t)^2,x(t), singsol=all)
\[ x \left (t \right ) = \frac {k}{1+{\mathrm e}^{-k t} c_{1} k} \]
✓ Solution by Mathematica
Time used: 0.963 (sec). Leaf size: 37
DSolve[x'[t]==k*x[t]-x[t]^2,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {k e^{k (t+c_1)}}{-1+e^{k (t+c_1)}} \\ x(t)\to 0 \\ x(t)\to k \\ \end{align*}