##### 4.3.27 $$y'(x)=a+b \sin (A x+B y(x))$$

ODE
$y'(x)=a+b \sin (A x+B y(x))$ ODE Classiﬁcation

[[_homogeneous, class C], _dAlembert]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.87412 (sec), leaf count = 94

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

Maple
cpu = 0.1 (sec), leaf count = 115

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

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

Mathematica raw output

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

Maple raw input

dsolve(diff(y(x),x) = a+b*sin(A*x+B*y(x)), y(x))

Maple raw output

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