Internal problem ID [13321]
Internal file name [OUTPUT/12493_Wednesday_February_14_2024_02_06_50_AM_75270919/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page
90
Problem number: 4.6 (d).
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]
\[ \boxed {y^{\prime }-\sin \left (y\right )=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{\sin \left (y \right )}d y &= x +c_{1}\\ \ln \left (\tan \left (\frac {y}{2}\right )\right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=2 \arctan \left ({\mathrm e}^{x +c_{1}}\right )\\ &=2 \arctan \left (c_{1} {\mathrm e}^{x}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \arctan \left (c_{1} {\mathrm e}^{x}\right ) \\ \end{align*}
Verification of solutions
\[ y = 2 \arctan \left (c_{1} {\mathrm e}^{x}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sin \left (y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\sin \left (y\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sin \left (y\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sin \left (y\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (-\cot \left (y\right )+\csc \left (y\right )\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arctan \left (\frac {2 \,{\mathrm e}^{x +c_{1}}}{\left ({\mathrm e}^{x +c_{1}}\right )^{2}+1}, -\frac {\left ({\mathrm e}^{x +c_{1}}\right )^{2}-1}{\left ({\mathrm e}^{x +c_{1}}\right )^{2}+1}\right ) \end {array} \]
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: 46
dsolve(diff(y(x),x)=sin(y(x)),y(x), singsol=all)
\[ y \left (x \right ) = \arctan \left (\frac {2 c_{1} {\mathrm e}^{x}}{c_{1}^{2} {\mathrm e}^{2 x}+1}, \frac {-c_{1}^{2} {\mathrm e}^{2 x}+1}{c_{1}^{2} {\mathrm e}^{2 x}+1}\right ) \]
✓ Solution by Mathematica
Time used: 0.293 (sec). Leaf size: 44
DSolve[y'[x]==Sin[y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\arccos (-\tanh (x+c_1)) \\ y(x)\to \arccos (-\tanh (x+c_1)) \\ y(x)\to 0 \\ y(x)\to -\pi \\ y(x)\to \pi \\ \end{align*}