Internal problem ID [12971]
Internal file name [OUTPUT/11624_Tuesday_November_07_2023_11_52_15_PM_92452613/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. Exercises section 1.6 page 89
Problem number: 9.
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]
\[ \boxed {y^{\prime }-\cos \left (y\right )=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{1+\cos \left (y \right )}d y &= t +c_{1}\\ \tan \left (\frac {y}{2}\right )&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=2 \arctan \left (t +c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \arctan \left (t +c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = 2 \arctan \left (t +c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\cos \left (y\right )=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=1+\cos \left (y\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{1+\cos \left (y\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{1+\cos \left (y\right )}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \tan \left (\frac {y}{2}\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=2 \arctan \left (t +c_{1} \right ) \end {array} \]
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: 0.016 (sec). Leaf size: 10
dsolve(diff(y(t),t)=1+cos(y(t)),y(t), singsol=all)
\[ y \left (t \right ) = 2 \arctan \left (t +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.462 (sec). Leaf size: 35
DSolve[y'[t]==1+cos[y[t]],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\cos (K[1])+1}dK[1]\&\right ][t+c_1] \\ y(t)\to \cos ^{(-1)}(-1) \\ \end{align*}