ODE
\[ y'(x)=\sqrt {a+b \cos (y(x))} \] ODE Classification
[_quadrature]
Book solution method
Separable ODE, Independent variable missing
Mathematica ✓
cpu = 0.115207 (sec), leaf count = 31
\[\left \{\left \{y(x)\to 2 \text {am}\left (\frac {1}{2} \sqrt {a+b} \left (x+c_1\right )|\frac {2 b}{a+b}\right )\right \}\right \}\]
Maple ✓
cpu = 0.011 (sec), leaf count = 21
\[ \left \{ x-\int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {a+b\cos \left ( {\it \_a} \right ) }}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[y'[x] == Sqrt[a + b*Cos[y[x]]],y[x],x]
Mathematica raw output
{{y[x] -> 2*JacobiAmplitude[(Sqrt[a + b]*(x + C[1]))/2, (2*b)/(a + b)]}}
Maple raw input
dsolve(diff(y(x),x) = (a+b*cos(y(x)))^(1/2), y(x),'implicit')
Maple raw output
x-Intat(1/(a+b*cos(_a))^(1/2),_a = y(x))+_C1 = 0