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

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

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

Mathematica ✓
cpu = 0.161276 (sec), leaf count = 71

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

Maple ✓
cpu = 0.01 (sec), leaf count = 62

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

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

Mathematica raw output

{{y[x] -> InverseFunction[((-2*a1*ArcTan[(a1 + 2*a2*#1)/Sqrt[-a1^2 + 4*a0*a2]])/
Sqrt[-a1^2 + 4*a0*a2] + Log[a0 + #1*(a1 + a2*#1)])/(2*a2) & ][x + C[1]]}}

Maple raw input

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

Maple raw output

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