Internal problem ID [11980]
Internal file name [OUTPUT/10633_Saturday_September_02_2023_02_48_46_PM_3774711/index.tex]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 7, Scalar autonomous ODEs. Exercises page 56
Problem number: 7.1 (iv).
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]
\[ \boxed {x^{\prime }+x \left (-x+1\right ) \left (2-x\right )=0} \]
Integrating both sides gives \begin {align*} \int -\frac {1}{x \left (x -2\right ) \left (x -1\right )}d x &= t +c_{1}\\ \ln \left (x -1\right )-\frac {\ln \left (x \right )}{2}-\frac {\ln \left (x -2\right )}{2}&=t +c_{1} \end {align*}
Solving for \(x\) gives these solutions \begin {align*} x_1&={\mathrm e}^{-\frac {\ln \left (-1+{\mathrm e}^{2 t +2 c_{1}}\right )}{2}+t +c_{1}}+1\\ &=\frac {{\mathrm e}^{t} c_{1}}{\sqrt {-1+{\mathrm e}^{2 t} c_{1}^{2}}}+1 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= \frac {{\mathrm e}^{t} c_{1}}{\sqrt {-1+{\mathrm e}^{2 t} c_{1}^{2}}}+1 \\ \end{align*}
Verification of solutions
\[ x = \frac {{\mathrm e}^{t} c_{1}}{\sqrt {-1+{\mathrm e}^{2 t} c_{1}^{2}}}+1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }+x \left (-x+1\right ) \left (2-x\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=-x \left (-x+1\right ) \left (2-x\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{x \left (-x+1\right ) \left (2-x\right )}=-1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{x \left (-x+1\right ) \left (2-x\right )}d t =\int \left (-1\right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (x-2\right )}{2}+\frac {\ln \left (x\right )}{2}-\ln \left (x-1\right )=-t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & \left \{x=\frac {\sqrt {-{\mathrm e}^{-2 t +2 c_{1}}+1}-1}{\sqrt {-{\mathrm e}^{-2 t +2 c_{1}}+1}}, x=\frac {\sqrt {-{\mathrm e}^{-2 t +2 c_{1}}+1}+1}{\sqrt {-{\mathrm e}^{-2 t +2 c_{1}}+1}}\right \} \end {array} \]
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: 0.079 (sec). Leaf size: 34
dsolve(diff(x(t),t)=-x(t)*(1-x(t))*(2-x(t)),x(t), singsol=all)
\[ x \left (t \right ) = \frac {c_{1} {\mathrm e}^{t}+\sqrt {-1+{\mathrm e}^{2 t} c_{1}^{2}}}{\sqrt {-1+{\mathrm e}^{2 t} c_{1}^{2}}} \]
✓ Solution by Mathematica
Time used: 19.885 (sec). Leaf size: 159
DSolve[x'[t]==-x[t]*(1-x[t])*(2-x[t]),x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {e^{2 t}-\sqrt {e^{4 t}+e^{2 (t+c_1)}}+e^{2 c_1}}{e^{2 t}+e^{2 c_1}} \\ x(t)\to \frac {e^{2 t}+\sqrt {e^{4 t}+e^{2 (t+c_1)}}+e^{2 c_1}}{e^{2 t}+e^{2 c_1}} \\ x(t)\to 0 \\ x(t)\to 1 \\ x(t)\to 2 \\ x(t)\to 1-e^{-2 t} \sqrt {e^{4 t}} \\ x(t)\to e^{-2 t} \sqrt {e^{4 t}}+1 \\ \end{align*}