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 = 0.386506 (sec), leaf count = 71
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {\log (\text {$\#$1} (\text {$\#$1} \text {a2}+\text {a1})+\text {a0})-\frac {2 \text {a1} \tan ^{-1}\left (\frac {2 \text {$\#$1} \text {a2}+\text {a1}}{\sqrt {4 \text {a0} \text {a2}-\text {a1}^2}}\right )}{\sqrt {4 \text {a0} \text {a2}-\text {a1}^2}}}{2 \text {a2}}\& \right ][x+c_1]\right \}\right \}\]
Maple ✓
cpu = 0.274 (sec), leaf count = 218
\[\left [y \left (x \right ) = \frac {4 \mathit {a0} \mathit {a2} \tan \left (\RootOf \left (2 \textit {\_C1} \mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}+2 x \mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}-\sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}\, \ln \left (\frac {4 \mathit {a0} \mathit {a2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-\mathit {a1}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}{4 \mathit {a2}}\right )+2 \textit {\_Z} \mathit {a1} \right )\right )-\mathit {a1}^{2} \tan \left (\RootOf \left (2 \textit {\_C1} \mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}+2 x \mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}-\sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}\, \ln \left (\frac {4 \mathit {a0} \mathit {a2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-\mathit {a1}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}{4 \mathit {a2}}\right )+2 \textit {\_Z} \mathit {a1} \right )\right )-\sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}\, \mathit {a1}}{2 \mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}}\right ]\] 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))
Maple raw output
[y(x) = 1/2*(4*a0*a2*tan(RootOf(2*_C1*a2*(4*a0*a2-a1^2)^(1/2)+2*x*a2*(4*a0*a2-a1^2)^(1/2)-(4*a0*a2-a1^2)^(1/2)*ln(1/4*(4*a0*a2*tan(_Z)^2-a1^2*tan(_Z)^2+4*a0*a2-a1^2)/a2)+2*_Z*a1))-a1^2*tan(RootOf(2*_C1*a2*(4*a0*a2-a1^2)^(1/2)+2*x*a2*(4*a0*a2-a1^2)^(1/2)-(4*a0*a2-a1^2)^(1/2)*ln(1/4*(4*a0*a2*tan(_Z)^2-a1^2*tan(_Z)^2+4*a0*a2-a1^2)/a2)+2*_Z*a1))-(4*a0*a2-a1^2)^(1/2)*a1)/a2/(4*a0*a2-a1^2)^(1/2)]