Internal problem ID [8094]
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.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2} \left (a \cos \relax (y)+b \right )-c \cos \relax (y)+d=0} \end {gather*}
✓ Solution by Maple
Time used: 0.043 (sec). Leaf size: 87
dsolve(diff(y(x),x)^2*(a*cos(y(x))+b)-c*cos(y(x))+d=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \arccos \left (\frac {d}{c}\right ) \\ x -\left (\int _{}^{y \relax (x )}\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 -\left (\int _{}^{y \relax (x )}-\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 \\ \end{align*}
✓ Solution by Mathematica
Time used: 13.043 (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 {$\#$1}}{2}\right ) \csc (\text {$\#$1}) \sqrt {a \cos (\text {$\#$1})+b} \sqrt {\frac {\cot ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d)}{c+d}} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (a+b) (d-c \cos (\text {$\#$1}))}{a d+b c}} \left (c (a+b) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right )|\frac {2 (b c+a d)}{(a+b) (c+d)}\right )+a (d-c) \Pi \left (\frac {b c+a d}{a c+b c};\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right )|\frac {2 (b c+a d)}{(a+b) (c+d)}\right )\right )}{c (a+b) \sqrt {c \cos (\text {$\#$1})-d} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d) (a \cos (\text {$\#$1})+b)}{a d+b c}}}\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [\frac {4 \sin ^2\left (\frac {\text {$\#$1}}{2}\right ) \csc (\text {$\#$1}) \sqrt {a \cos (\text {$\#$1})+b} \sqrt {\frac {\cot ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d)}{c+d}} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (a+b) (d-c \cos (\text {$\#$1}))}{a d+b c}} \left (c (a+b) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right )|\frac {2 (b c+a d)}{(a+b) (c+d)}\right )+a (d-c) \Pi \left (\frac {b c+a d}{a c+b c};\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right )|\frac {2 (b c+a d)}{(a+b) (c+d)}\right )\right )}{c (a+b) \sqrt {c \cos (\text {$\#$1})-d} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d) (a \cos (\text {$\#$1})+b)}{a d+b c}}}\&\right ][x+c_1] \\ y(x)\to -\text {ArcCos}\left (\frac {d}{c}\right ) \\ y(x)\to \text {ArcCos}\left (\frac {d}{c}\right ) \\ \end{align*}