Internal problem ID [14941]
Internal file name [OUTPUT/14951_Monday_April_15_2024_12_04_15_AM_13746180/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 1. Basic concepts and definitions. Exercises page 18
Problem number: 10.
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]
\[ \boxed {y^{\prime }+\cot \left (y\right )=1} \]
Integrating both sides gives \begin{align*} \int \frac {1}{1-\cot \left (y \right )}d y &= \int d x \\ \frac {y}{2}+\frac {\ln \left (\tan \left (y\right )-1\right )}{2}-\frac {\ln \left (\sec \left (y\right )^{2}\right )}{4}&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \frac {y}{2}+\frac {\ln \left (\tan \left (y\right )-1\right )}{2}-\frac {\ln \left (\sec \left (y\right )^{2}\right )}{4} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \frac {y}{2}+\frac {\ln \left (\tan \left (y\right )-1\right )}{2}-\frac {\ln \left (\sec \left (y\right )^{2}\right )}{4} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\cot \left (y\right )=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=1-\cot \left (y\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{1-\cot \left (y\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{1-\cot \left (y\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (-1+\cot \left (y\right )\right )}{2}-\frac {\ln \left (\cot \left (y\right )^{2}+1\right )}{4}-\frac {\pi }{4}+\frac {y}{2}=x +c_{1} \end {array} \]
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: 0.14 (sec). Leaf size: 29
dsolve(diff(y(x),x)=1-cot(y(x)),y(x), singsol=all)
\[ x +\frac {\ln \left (\csc \left (y \left (x \right )\right )^{2}\right )}{4}+\frac {\pi }{4}-\frac {\ln \left (-1+\cot \left (y \left (x \right )\right )\right )}{2}-\frac {y \left (x \right )}{2}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.376 (sec). Leaf size: 69
DSolve[y'[x]==1-Cot[y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\left (\frac {1}{4}+\frac {i}{4}\right ) \log (-\tan (\text {$\#$1})+i)-\frac {1}{2} \log (1-\tan (\text {$\#$1}))+\left (\frac {1}{4}-\frac {i}{4}\right ) \log (\tan (\text {$\#$1})+i)\&\right ][-x+c_1] \\ y(x)\to \frac {\pi }{4} \\ \end{align*}