##### 4.3.38 $$y'(x)=f(a+b x+c y(x))$$

ODE
$y'(x)=f(a+b x+c y(x))$ ODE Classiﬁcation

[[_homogeneous, class C], _dAlembert]

Book solution method
Equation linear in the variables, $$y'(x)=f(a+b x+ c y(x))$$

Mathematica
cpu = 0.388896 (sec), leaf count = 116

$\text {Solve}\left [c_1=\int _1^{y(x)}\left (-\frac {c}{b+c f(a+b x+c K[2])}-\int _1^x\frac {b c^2 f'(a+b K[1]+c K[2])}{(b+c f(a+b K[1]+c K[2]))^2}dK[1]\right )dK[2]+\int _1^x\frac {c f(a+b K[1]+c y(x))}{b+c f(a+b K[1]+c y(x))}dK[1],y(x)\right ]$

Maple
cpu = 0.047 (sec), leaf count = 39

$\left [y \left (x \right ) = \frac {\RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {1}{f \left (\textit {\_a} c +a \right ) c +b}d \textit {\_a} \right ) c -x +\textit {\_C1} \right ) c -b x}{c}\right ]$ Mathematica raw input

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

Mathematica raw output

Solve[C[1] == Inactive[Integrate][(c*f[a + b*K[1] + c*y[x]])/(b + c*f[a + b*K[1]
 + c*y[x]]), {K[1], 1, x}] + Inactive[Integrate][-(c/(b + c*f[a + b*x + c*K[2]])
) - Inactive[Integrate][(b*c^2*Derivative[1][f][a + b*K[1] + c*K[2]])/(b + c*f[a
 + b*K[1] + c*K[2]])^2, {K[1], 1, x}], {K[2], 1, y[x]}], y[x]]

Maple raw input

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

Maple raw output

[y(x) = (RootOf(Intat(1/(f(_a*c+a)*c+b),_a = _Z)*c-x+_C1)*c-b*x)/c]