Internal problem ID [3141]
Internal file name [OUTPUT/2633_Sunday_June_05_2022_08_37_45_AM_26296455/index.tex]
Book: An introduction to the solution and applications of differential equations, J.W. Searl,
1966
Section: Chapter 4, Ex. 4.2
Problem number: 2.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {x \left (x -1\right ) y^{\prime }-\cot \left (y\right )=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {\cot \left (y \right )}{x \left (x -1\right )} \end {align*}
Where \(f(x)=\frac {1}{x \left (x -1\right )}\) and \(g(y)=\cot \left (y \right )\). Integrating both sides gives \begin{align*} \frac {1}{\cot \left (y \right )} \,dy &= \frac {1}{x \left (x -1\right )} \,d x \\ \int { \frac {1}{\cot \left (y \right )} \,dy} &= \int {\frac {1}{x \left (x -1\right )} \,d x} \\ -\ln \left (\cos \left (y \right )\right )&=\ln \left (x -1\right )-\ln \left (x \right )+c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} \frac {1}{\cos \left (y \right )} &= {\mathrm e}^{\ln \left (x -1\right )-\ln \left (x \right )+c_{1}} \end {align*}
Which simplifies to \begin {align*} \sec \left (y \right ) &= c_{2} {\mathrm e}^{\ln \left (x -1\right )-\ln \left (x \right )} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \operatorname {arcsec}\left (\frac {c_{2} {\mathrm e}^{c_{1}} \left (x -1\right )}{x}\right ) \\ \end{align*}
Verification of solutions
\[ y = \operatorname {arcsec}\left (\frac {c_{2} {\mathrm e}^{c_{1}} \left (x -1\right )}{x}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (x -1\right ) y^{\prime }-\cot \left (y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\cot \left (y\right )}{x \left (x -1\right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\cot \left (y\right )}=\frac {1}{x \left (x -1\right )} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\cot \left (y\right )}d x =\int \frac {1}{x \left (x -1\right )}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (\cos \left (y\right )\right )=\ln \left (x -1\right )-\ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arccos \left (\frac {x}{{\mathrm e}^{c_{1}} \left (x -1\right )}\right ) \end {array} \]
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: 0.015 (sec). Leaf size: 15
dsolve(x*(x-1)*diff(y(x),x)=cot(y(x)),y(x), singsol=all)
\[ y \left (x \right ) = \arccos \left (\frac {x}{c_{1} \left (x -1\right )}\right ) \]
✓ Solution by Mathematica
Time used: 52.823 (sec). Leaf size: 59
DSolve[x*(x-1)*y'[x]==Cot[y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\arccos \left (-\frac {e^{-c_1} x}{x-1}\right ) \\ y(x)\to \arccos \left (-\frac {e^{-c_1} x}{x-1}\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}