4.3.11 \(y'(x)=a+b \cos (A x+B y(x))\)

ODE
\[ y'(x)=a+b \cos (A x+B y(x)) \] ODE Classification

[[_homogeneous, `class C`], _dAlembert]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.627935 (sec), leaf count = 102

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

Maple
cpu = 0.069 (sec), leaf count = 84

\[ \left \{ x-2\,{\frac {1}{\sqrt {{B}^{2}{a}^{2}-{B}^{2}{b}^{2}+2\,ABa+{A}^{2}}}\arctan \left ({\frac { \left (A+ \left (a-b \right ) B \right ) \tan \left (1/2\,Ax+1/2\,By \relax (x ) \right ) }{\sqrt {{B}^{2}{a}^{2}-{B}^{2}{b}^{2}+2\,ABa+{A}^{2}}}} \right ) }-{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

dsolve(diff(y(x),x) = a+b*cos(A*x+B*y(x)), y(x),'implicit')

Maple raw output

x-2/(B^2*a^2-B^2*b^2+2*A*B*a+A^2)^(1/2)*arctan((A+(a-b)*B)/(B^2*a^2-B^2*b^2+2*A*
B*a+A^2)^(1/2)*tan(1/2*A*x+1/2*B*y(x)))-_C1 = 0