Internal problem ID [8413]
Internal file name [OUTPUT/7346_Sunday_June_05_2022_10_51_21_PM_61266139/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 76.
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]
\[ \boxed {y^{\prime }-a \cos \left (y\right )=-b} \]
Integrating both sides gives \begin {align*} \int \frac {1}{a \cos \left (y \right )-b}d y &= x +c_{1}\\ \frac {2 \,\operatorname {arctanh}\left (\frac {\left (a +b \right ) \tan \left (\frac {y}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=2 \arctan \left (\frac {\tanh \left (\frac {c_{1} \sqrt {\left (a -b \right ) \left (a +b \right )}}{2}+\frac {x \sqrt {\left (a -b \right ) \left (a +b \right )}}{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}{a +b}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \arctan \left (\frac {\tanh \left (\frac {c_{1} \sqrt {\left (a -b \right ) \left (a +b \right )}}{2}+\frac {x \sqrt {\left (a -b \right ) \left (a +b \right )}}{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}{a +b}\right ) \\ \end{align*}
Verification of solutions
\[ y = 2 \arctan \left (\frac {\tanh \left (\frac {c_{1} \sqrt {\left (a -b \right ) \left (a +b \right )}}{2}+\frac {x \sqrt {\left (a -b \right ) \left (a +b \right )}}{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}{a +b}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a \cos \left (y\right )=-b \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \cos \left (y\right )-b \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{a \cos \left (y\right )-b}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{a \cos \left (y\right )-b}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {2 \,\mathrm {arctanh}\left (\frac {\tan \left (\frac {y}{2}\right ) \left (a +b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=2 \arctan \left (\frac {\tanh \left (\frac {c_{1} \sqrt {\left (a -b \right ) \left (a +b \right )}}{2}+\frac {x \sqrt {\left (a -b \right ) \left (a +b \right )}}{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}{a +b}\right ) \end {array} \]
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: 41
dsolve(diff(y(x),x) - a*cos(y(x)) + b=0,y(x), singsol=all)
\[ y \left (x \right ) = 2 \arctan \left (\frac {\tanh \left (\frac {\sqrt {a^{2}-b^{2}}\, \left (x +c_{1} \right )}{2}\right ) \sqrt {a^{2}-b^{2}}}{a +b}\right ) \]
✓ Solution by Mathematica
Time used: 60.136 (sec). Leaf size: 51
DSolve[y'[x] - a*Cos[y[x]] + b==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2 \arctan \left (\frac {(a-b) \tanh \left (\frac {1}{2} \sqrt {a^2-b^2} (x-c_1)\right )}{\sqrt {a^2-b^2}}\right ) \]