Internal problem ID [11981]
Internal file name [OUTPUT/10634_Saturday_September_02_2023_02_48_47_PM_21281142/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 (v).
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]
\[ \boxed {x^{\prime }-x^{2}+x^{4}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-x^{4}+x^{2}}d x &= \int {dt}\\ \int _{}^{x}\frac {1}{-\textit {\_a}^{4}+\textit {\_a}^{2}}d \textit {\_a}&= t +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{x}\frac {1}{-\textit {\_a}^{4}+\textit {\_a}^{2}}d \textit {\_a} &= t +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{x}\frac {1}{-\textit {\_a}^{4}+\textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }-x^{2}+x^{4}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=x^{2}-x^{4} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{x^{2}-x^{4}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{x^{2}-x^{4}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (x+1\right )}{2}-\frac {1}{x}-\frac {\ln \left (x-1\right )}{2}=t +c_{1} \end {array} \]
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: 0.078 (sec). Leaf size: 47
dsolve(diff(x(t),t)=x(t)^2-x(t)^4,x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}-2\right ) {\mathrm e}^{\textit {\_Z}}+2 c_{1} {\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 t \,{\mathrm e}^{\textit {\_Z}}-\ln \left ({\mathrm e}^{\textit {\_Z}}-2\right )-2 c_{1} +\textit {\_Z} -2 t +2\right )}-1 \]
✓ Solution by Mathematica
Time used: 0.414 (sec). Leaf size: 53
DSolve[x'[t]==x[t]^2-x[t]^4,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \text {InverseFunction}\left [\frac {1}{\text {$\#$1}}+\frac {1}{2} \log (1-\text {$\#$1})-\frac {1}{2} \log (\text {$\#$1}+1)\&\right ][-t+c_1] \\ x(t)\to -1 \\ x(t)\to 0 \\ x(t)\to 1 \\ \end{align*}