##### 4.3.14 $$y'(x)=a+b \cos (y(x))$$

ODE
$y'(x)=a+b \cos (y(x))$ ODE Classiﬁcation

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

Mathematica
cpu = 0.297486 (sec), leaf count = 47

$\left \{\left \{y(x)\to 2 \tan ^{-1}\left (\frac {(a+b) \tanh \left (\frac {1}{2} \sqrt {b^2-a^2} (x+c_1)\right )}{\sqrt {b^2-a^2}}\right )\right \}\right \}$

Maple
cpu = 0.043 (sec), leaf count = 56

$\left [y \left (x \right ) = 2 \arctan \left (\frac {\tan \left (\frac {\textit {\_C1} \sqrt {\left (a +b \right ) \left (a -b \right )}}{2}+\frac {x \sqrt {\left (a +b \right ) \left (a -b \right )}}{2}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}{a -b}\right )\right ]$ Mathematica raw input

DSolve[y'[x] == a + b*Cos[y[x]],y[x],x]

Mathematica raw output

{{y[x] -> 2*ArcTan[((a + b)*Tanh[(Sqrt[-a^2 + b^2]*(x + C[1]))/2])/Sqrt[-a^2 + b
^2]]}}

Maple raw input

dsolve(diff(y(x),x) = a+b*cos(y(x)), y(x))

Maple raw output

[y(x) = 2*arctan(tan(1/2*_C1*((a+b)*(a-b))^(1/2)+1/2*x*((a+b)*(a-b))^(1/2))*((a+
b)*(a-b))^(1/2)/(a-b))]