Internal problem ID [11979]
Internal file name [OUTPUT/10632_Saturday_September_02_2023_02_48_42_PM_26439748/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 (iii).
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]
\[ \boxed {x^{\prime }-\left (x+1\right ) \left (2-x\right ) \sin \left (x\right )=0} \]
Integrating both sides gives \begin {align*} \int _{}^{x}-\frac {1}{\sin \left (\textit {\_a} \right ) \left (\textit {\_a} +1\right ) \left (\textit {\_a} -2\right )}d \textit {\_a} = t +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{x}-\frac {1}{\sin \left (\textit {\_a} \right ) \left (\textit {\_a} +1\right ) \left (\textit {\_a} -2\right )}d \textit {\_a} &= t +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{x}-\frac {1}{\sin \left (\textit {\_a} \right ) \left (\textit {\_a} +1\right ) \left (\textit {\_a} -2\right )}d \textit {\_a} = t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }-\left (x+1\right ) \left (2-x\right ) \sin \left (x\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=\left (x+1\right ) \left (2-x\right ) \sin \left (x\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{\left (x+1\right ) \left (2-x\right ) \sin \left (x\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{\left (x+1\right ) \left (2-x\right ) \sin \left (x\right )}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {x^{\prime }}{\left (x+1\right ) \left (2-x\right ) \sin \left (x\right )}d t =t +c_{1} \end {array} \]
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: 0.0 (sec). Leaf size: 24
dsolve(diff(x(t),t)=(1+x(t))*(2-x(t))*sin(x(t)),x(t), singsol=all)
\[ t +\int _{}^{x \left (t \right )}\frac {\csc \left (\textit {\_a} \right )}{\left (\textit {\_a} +1\right ) \left (\textit {\_a} -2\right )}d \textit {\_a} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 15.593 (sec). Leaf size: 52
DSolve[x'[t]==(1+x[t])*(2-x[t])*Sin[x[t]],x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\csc (K[1])}{(K[1]-2) (K[1]+1)}dK[1]\&\right ][-t+c_1] \\ x(t)\to -1 \\ x(t)\to 0 \\ x(t)\to 2 \\ \end{align*}