Internal problem ID [14081]
Internal file name [OUTPUT/13762_Saturday_March_02_2024_02_49_19_PM_48254115/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number: 45.
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]
\[ \boxed {y^{\prime }=\cos \left (x \right ) \cot \left (x \right )} \]
Integrating both sides gives \begin {align*} y &= \int { \cos \left (x \right ) \cot \left (x \right )\,\mathop {\mathrm {d}x}}\\ &= \cos \left (x \right )+\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \cos \left (x \right )+\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \cos \left (x \right )+\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\cos \left (x \right ) \cot \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \cos \left (x \right ) \cot \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\cos \left (x \right )+\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\cos \left (x \right )+\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 16
dsolve(diff(y(x),x)=cos(x)*cot(x),y(x), singsol=all)
\[ y \left (x \right ) = \cos \left (x \right )+\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 26
DSolve[y'[x]==Cos[x]*Cot[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \cos (x)+\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right )+c_1 \]