4.2.27 \(y'(x)=f(x) \left (a+b y(x)+c y(x)^2\right )\)

ODE
\[ y'(x)=f(x) \left (a+b y(x)+c y(x)^2\right ) \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.304564 (sec), leaf count = 63

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

Maple ✓
cpu = 0.046 (sec), leaf count = 60

\[\left [y \left (x \right ) = -\frac {-\tan \left (\frac {\left (\int f \left (x \right )d x \right ) \sqrt {4 c a -b^{2}}}{2}+\frac {\textit {\_C1} \sqrt {4 c a -b^{2}}}{2}\right ) \sqrt {4 c a -b^{2}}+b}{2 c}\right ]\] 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] + Inactive[Inte
grate][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))

Maple raw output

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