Internal problem ID [7940]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 360.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {y^{\prime } \cos \left (a y\right )-b \left (1-c \cos \left (a y\right )\right ) \sqrt {\cos ^{2}\left (a y\right )-1+c \cos \left (a y\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 48
dsolve(diff(y(x),x)*cos(a*y(x))-b*(1-c*cos(a*y(x)))*(cos(a*y(x))^2-1+c*cos(a*y(x)))^(1/2) = 0,y(x), singsol=all)
\[ c_{1}+x +\int _{}^{y \relax (x )}\frac {2 \cos \left (a \textit {\_a} \right )}{b \left (c \cos \left (a \textit {\_a} \right )-1\right ) \sqrt {-2+2 \cos \left (2 a \textit {\_a} \right )+4 c \cos \left (a \textit {\_a} \right )}}d \textit {\_a} = 0 \]
✓ Solution by Mathematica
Time used: 25.488 (sec). Leaf size: 500
DSolve[-(b*(1 - c*Cos[a*y[x]])*Sqrt[-1 + c*Cos[a*y[x]] + Cos[a*y[x]]^2]) + Cos[a*y[x]]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {i (\cos (\text {$\#$1} a)+1) \sqrt {\frac {2 c \cos (\text {$\#$1} a)+\cos (2 \text {$\#$1} a)-1}{(\cos (\text {$\#$1} a)+1)^2}} \sqrt {\frac {c \tan ^2\left (\frac {\text {$\#$1} a}{2}\right )+\sqrt {c^2+4}+2}{\sqrt {c^2+4}+2}} \sqrt {1-\frac {c \tan ^2\left (\frac {\text {$\#$1} a}{2}\right )}{\sqrt {c^2+4}-2}} \left ((c-1) \text {EllipticF}\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{\sqrt {c^2+4}-2}} \tan \left (\frac {\text {$\#$1} a}{2}\right )\right ),\frac {2-\sqrt {c^2+4}}{\sqrt {c^2+4}+2}\right )+2 \Pi \left (\frac {(c+1) \left (\sqrt {c^2+4}-2\right )}{(c-1) c};i \sinh ^{-1}\left (\sqrt {-\frac {c}{\sqrt {c^2+4}-2}} \tan \left (\frac {a \text {$\#$1}}{2}\right )\right )|\frac {2-\sqrt {c^2+4}}{\sqrt {c^2+4}+2}\right )\right )}{a \left (c^2-1\right ) \sqrt {\frac {c}{4-2 \sqrt {c^2+4}}} \sqrt {2 c \cos (\text {$\#$1} a)+\cos (2 \text {$\#$1} a)-1} \sqrt {-c \tan ^4\left (\frac {\text {$\#$1} a}{2}\right )-4 \tan ^2\left (\frac {\text {$\#$1} a}{2}\right )+c}}\&\right ]\left [-\frac {b x}{\sqrt {2}}+c_1\right ] \\ y(x)\to -\frac {\sec ^{-1}(c)}{a} \\ y(x)\to \frac {\sec ^{-1}(c)}{a} \\ y(x)\to -\frac {\text {ArcCos}\left (\frac {1}{2} \left (-\sqrt {c^2+4}-c\right )\right )}{a} \\ y(x)\to \frac {\text {ArcCos}\left (\frac {1}{2} \left (-\sqrt {c^2+4}-c\right )\right )}{a} \\ y(x)\to -\frac {\text {ArcCos}\left (\frac {1}{2} \left (\sqrt {c^2+4}-c\right )\right )}{a} \\ y(x)\to \frac {\text {ArcCos}\left (\frac {1}{2} \left (\sqrt {c^2+4}-c\right )\right )}{a} \\ \end{align*}