4.2.10 \(y'(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2\)

ODE
\[ y'(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2 \] ODE Classification

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

Mathematica
cpu = 0.0369065 (sec), leaf count = 54

\[\left \{\left \{y(x)\to \frac {\sqrt {4 \text {a0} \text {a2}-\text {a1}^2} \tan \left (\frac {1}{2} \left (c_1+x\right ) \sqrt {4 \text {a0} \text {a2}-\text {a1}^2}\right )-\text {a1}}{2 \text {a2}}\right \}\right \}\]

Maple
cpu = 0.012 (sec), leaf count = 41

\[ \left \{ x-2\,{\frac {1}{\sqrt {4\,{\it a0}\,{\it a2}-{{\it a1}}^{2}}}\arctan \left ({\frac {2\,{\it a2}\,y \relax (x ) +{\it a1}}{\sqrt {4\,{\it a0}\,{\it a2}-{{\it a1}}^{2}}}} \right ) }+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y'[x] == a0 + a1*y[x] + a2*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> (-a1 + Sqrt[-a1^2 + 4*a0*a2]*Tan[(Sqrt[-a1^2 + 4*a0*a2]*(x + C[1]))/2]
)/(2*a2)}}

Maple raw input

dsolve(diff(y(x),x) = a0+a1*y(x)+a2*y(x)^2, y(x),'implicit')

Maple raw output

x-2/(4*a0*a2-a1^2)^(1/2)*arctan((2*a2*y(x)+a1)/(4*a0*a2-a1^2)^(1/2))+_C1 = 0