4.2.27 $y^{\prime }(x)=f(x)(a+by(x)+cy(x{)}^2)$

ODE
$$y^{\prime }(x)=f(x)(a+by(x)+cy(x{)}^2)$$ ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.066664 (sec), leaf count = 62

$$\left\{\{y(x)\to \frac{\sqrt{4ac-b^2}\tan \left(\frac{1}{2}\sqrt{4ac-b^2}({\int }_1^{x}f(K[1])dK[1]+c_1)\right)-b}{2c}\}\right\}$$

Maple ✓
cpu = 0.011 (sec), leaf count = 44

$$\{\int f\left(x\right)\mathrm{d}x-2\frac{1}{\sqrt{4ca-b^2}}\arctan \left(\frac{2cy\left(x\right)+b}{\sqrt{4ca-b^2}}\right)+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[y'[x] == f[x]*(a + b*y[x] + c*y[x]^2),y[x],x]

Mathematica raw output

{{y[x] -> (-b + Sqrt[-b^2 + 4*a*c]*Tan[(Sqrt[-b^2 + 4*a*c]*(C[1] + Integrate[f[K
[1]], {K[1], 1, x}]))/2])/(2*c)}}

Maple raw input

dsolve(diff(y(x),x) = (a+b*y(x)+c*y(x)^2)*f(x), y(x),'implicit')

Maple raw output

Int(f(x),x)-2/(4*a*c-b^2)^(1/2)*arctan((2*c*y(x)+b)/(4*a*c-b^2)^(1/2))+_C1 = 0