Internal problem ID [13032]
Internal file name [OUTPUT/11685_Wednesday_November_08_2023_03_28_45_AM_58753678/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Review Exercises for chapter 1. page
136
Problem number: 6.
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]
\[ \boxed {y^{\prime }-\sin \left (y\right )^{2}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{\sin \left (y \right )^{2}}d y &= t +c_{1}\\ -\cot \left (y \right )&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\pi -\operatorname {arccot}\left (t +c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \pi -\operatorname {arccot}\left (t +c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \pi -\operatorname {arccot}\left (t +c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sin \left (y\right )^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\sin \left (y\right )^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sin \left (y\right )^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{\sin \left (y\right )^{2}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\cot \left (y\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\pi -\mathrm {arccot}\left (t +c_{1} \right ) \end {array} \]
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: 0.015 (sec). Leaf size: 12
dsolve(diff(y(t),t)=sin(y(t))^2,y(t), singsol=all)
\[ y \left (t \right ) = \frac {\pi }{2}+\arctan \left (t +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.319 (sec). Leaf size: 19
DSolve[y'[t]==Sin[y[t]]^2,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -\cot ^{-1}(t-2 c_1) \\ y(t)\to 0 \\ \end{align*}