Internal problem ID [8848]
Internal file name [OUTPUT/7783_Monday_June_06_2022_12_19_42_AM_51764964/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 514.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {{y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )=-d} \] Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}{a \cos \left (y\right )+b} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}{a \cos \left (y\right )+b} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int \frac {a \cos \left (y \right )+b}{\sqrt {\left (a \cos \left (y \right )+b \right ) \left (c \cos \left (y \right )-d \right )}}d y &= \int {dx}\\ \int _{}^{y}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a}&= x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} = x +c_{1} \] Verified OK.
Solving equation (2)
Integrating both sides gives \begin {align*} \int -\frac {a \cos \left (y \right )+b}{\sqrt {\left (a \cos \left (y \right )+b \right ) \left (c \cos \left (y \right )-d \right )}}d y &= \int {dx}\\ \int _{}^{y}-\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a}&= x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}-\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}-\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} = x +c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )=-d \\ \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} \\ {} & {} & \left [y^{\prime }=\frac {\sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}{a \cos \left (y\right )+b}, y^{\prime }=-\frac {\sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}{a \cos \left (y\right )+b}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}{a \cos \left (y\right )+b} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\left (a \cos \left (y\right )+b \right ) y^{\prime }}{\sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\left (a \cos \left (y\right )+b \right ) y^{\prime }}{\sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {2 \sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right ) \sin \left (y\right )^{2}}\, \left (a +b \right ) \sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}\, \left (\cos \left (y\right )+1\right )^{2} \sqrt {\frac {c \cos \left (y\right )-d}{\left (-d +c \right ) \left (\cos \left (y\right )+1\right )}}\, \sqrt {\frac {a \cos \left (y\right )+b}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}\, \left (a \mathit {EllipticF}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )-b \mathit {EllipticF}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )-2 a \mathit {EllipticPi}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, -\frac {a +b}{a -b}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )\right )}{\left (a -b \right ) \sqrt {-\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right ) \left (\cos \left (y\right )-1\right ) \left (\cos \left (y\right )+1\right )}\, \sin \left (y\right ) \sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}=x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}{a \cos \left (y\right )+b} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\left (a \cos \left (y\right )+b \right ) y^{\prime }}{\sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}=-1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\left (a \cos \left (y\right )+b \right ) y^{\prime }}{\sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}d x =\int \left (-1\right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {2 \sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right ) \sin \left (y\right )^{2}}\, \left (a +b \right ) \sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}\, \left (\cos \left (y\right )+1\right )^{2} \sqrt {\frac {c \cos \left (y\right )-d}{\left (-d +c \right ) \left (\cos \left (y\right )+1\right )}}\, \sqrt {\frac {a \cos \left (y\right )+b}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}\, \left (a \mathit {EllipticF}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )-b \mathit {EllipticF}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )-2 a \mathit {EllipticPi}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, -\frac {a +b}{a -b}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )\right )}{\left (a -b \right ) \sqrt {-\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right ) \left (\cos \left (y\right )-1\right ) \left (\cos \left (y\right )+1\right )}\, \sin \left (y\right ) \sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}=-x +c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{-\frac {2 \sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right ) \sin \left (y\right )^{2}}\, \left (a +b \right ) \sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}\, \left (\cos \left (y\right )+1\right )^{2} \sqrt {\frac {c \cos \left (y\right )-d}{\left (-d +c \right ) \left (\cos \left (y\right )+1\right )}}\, \sqrt {\frac {a \cos \left (y\right )+b}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}\, \left (a \mathit {EllipticF}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )-b \mathit {EllipticF}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )-2 a \mathit {EllipticPi}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, -\frac {a +b}{a -b}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )\right )}{\left (a -b \right ) \sqrt {-\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right ) \left (\cos \left (y\right )-1\right ) \left (\cos \left (y\right )+1\right )}\, \sin \left (y\right ) \sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}=-x +c_{1} , -\frac {2 \sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right ) \sin \left (y\right )^{2}}\, \left (a +b \right ) \sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}\, \left (\cos \left (y\right )+1\right )^{2} \sqrt {\frac {c \cos \left (y\right )-d}{\left (-d +c \right ) \left (\cos \left (y\right )+1\right )}}\, \sqrt {\frac {a \cos \left (y\right )+b}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}\, \left (a \mathit {EllipticF}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )-b \mathit {EllipticF}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )-2 a \mathit {EllipticPi}\left (\sqrt {-\frac {\left (a -b \right ) \left (\cos \left (y\right )-1\right )}{\left (a +b \right ) \left (\cos \left (y\right )+1\right )}}, -\frac {a +b}{a -b}, \sqrt {\frac {\left (d +c \right ) \left (a +b \right )}{\left (-d +c \right ) \left (a -b \right )}}\right )\right )}{\left (a -b \right ) \sqrt {-\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right ) \left (\cos \left (y\right )-1\right ) \left (\cos \left (y\right )+1\right )}\, \sin \left (y\right ) \sqrt {\left (a \cos \left (y\right )+b \right ) \left (c \cos \left (y\right )-d \right )}}=x +c_{1} \right \} \end {array} \]
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables <- differential order: 1; missing x successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 84
dsolve(diff(y(x),x)^2*(a*cos(y(x))+b)-c*cos(y(x))+d=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \arccos \left (\frac {d}{c}\right ) \\ x -\left (\int _{}^{y \left (x \right )}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} \right )-c_{1} &= 0 \\ x +\int _{}^{y \left (x \right )}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} -c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.351 (sec). Leaf size: 627
DSolve[d - c*Cos[y[x]] + (b + a*Cos[y[x]])*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {4 \sin ^2\left (\frac {\text {$\#$1}}{2}\right ) \csc (\text {$\#$1}) \sqrt {a \cos (\text {$\#$1})+b} \sqrt {\frac {\cot ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d)}{c+d}} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (a+b) (d-c \cos (\text {$\#$1}))}{a d+b c}} \left (c (a+b) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c+a d)}{(a+b) (c+d)}\right )+a (d-c) \operatorname {EllipticPi}\left (\frac {b c+a d}{a c+b c},\arcsin \left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c+a d)}{(a+b) (c+d)}\right )\right )}{c (a+b) \sqrt {c \cos (\text {$\#$1})-d} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d) (a \cos (\text {$\#$1})+b)}{a d+b c}}}\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [\frac {4 \sin ^2\left (\frac {\text {$\#$1}}{2}\right ) \csc (\text {$\#$1}) \sqrt {a \cos (\text {$\#$1})+b} \sqrt {\frac {\cot ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d)}{c+d}} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (a+b) (d-c \cos (\text {$\#$1}))}{a d+b c}} \left (c (a+b) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c+a d)}{(a+b) (c+d)}\right )+a (d-c) \operatorname {EllipticPi}\left (\frac {b c+a d}{a c+b c},\arcsin \left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c+a d)}{(a+b) (c+d)}\right )\right )}{c (a+b) \sqrt {c \cos (\text {$\#$1})-d} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d) (a \cos (\text {$\#$1})+b)}{a d+b c}}}\&\right ][x+c_1] \\ y(x)\to -\arccos \left (\frac {d}{c}\right ) \\ y(x)\to \arccos \left (\frac {d}{c}\right ) \\ \end{align*}