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 = 0.780987 (sec), leaf count = 94

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

Maple
cpu = 0.072 (sec), leaf count = 85

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

DSolve[y'[x] == a + b*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),'implicit')

Maple raw output

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