Internal problem ID [13327]
Internal file name [OUTPUT/12499_Wednesday_February_14_2024_11_52_49_PM_55840271/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: 4.7 (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]
\[ \boxed {y^{\prime }-\tan \left (y\right )=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{\tan \left (y \right )}d y &= x +c_{1}\\ \ln \left (\sin \left (y \right )\right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\arcsin \left ({\mathrm e}^{x +c_{1}}\right )\\ &=\arcsin \left (c_{1} {\mathrm e}^{x}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \arcsin \left (c_{1} {\mathrm e}^{x}\right ) \\ \end{align*}
Verification of solutions
\[ y = \arcsin \left (c_{1} {\mathrm e}^{x}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\tan \left (y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\tan \left (y\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\tan \left (y\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\tan \left (y\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (\sin \left (y\right )\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arcsin \left ({\mathrm e}^{x +c_{1}}\right ) \end {array} \]
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: 0.0 (sec). Leaf size: 9
dsolve(diff(y(x),x)=tan(y(x)),y(x), singsol=all)
\[ y \left (x \right ) = \arcsin \left (c_{1} {\mathrm e}^{x}\right ) \]
✓ Solution by Mathematica
Time used: 40.257 (sec). Leaf size: 17
DSolve[y'[x]==Tan[y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \arcsin \left (e^{x+c_1}\right ) \\ y(x)\to 0 \\ \end{align*}