4.2.35 \(y'(x)=y(x)^2 (a+b x y(x))\)

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

[[_homogeneous, `class G`], _Abel]

Book solution method
Abel ODE, First kind

Mathematica
cpu = 0.144999 (sec), leaf count = 94

\[\text {Solve}\left [\frac {a^2 \log \left (\frac {b x y(x) (a+b x y(x))+b}{b^2 x^2 y(x)^2}\right )+\frac {2 a^2 \tan ^{-1}\left (\frac {a+2 b x y(x)}{a \sqrt {\frac {4 b}{a^2}-1}}\right )}{\sqrt {\frac {4 b}{a^2}-1}}+2 a^2 \log (x)+2 b c_1}{b}=0,y(x)\right ]\]

Maple
cpu = 0.049 (sec), leaf count = 79

\[ \left \{ -{1 \left (a{\it Artanh} \left ({(2\,bxy \relax (x ) +a){\frac {1}{\sqrt {{a}^{2}-4\,b}}}} \right ) -{\frac {\ln \left (b{x}^{2} \left (y \relax (x ) \right ) ^{2}+axy \relax (x ) +1 \right ) -2\,{\it \_C1}+2\,\ln \relax (x ) -2\,\ln \left (xy \relax (x ) \right ) }{2}\sqrt {{a}^{2}-4\,b}} \right ) {\frac {1}{\sqrt {{a}^{2}-4\,b}}}}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

Solve[((2*a^2*ArcTan[(a + 2*b*x*y[x])/(a*Sqrt[-1 + (4*b)/a^2])])/Sqrt[-1 + (4*b)
/a^2] + 2*b*C[1] + 2*a^2*Log[x] + a^2*Log[(b + b*x*y[x]*(a + b*x*y[x]))/(b^2*x^2
*y[x]^2)])/b == 0, y[x]]

Maple raw input

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

Maple raw output

-(a*arctanh((2*b*x*y(x)+a)/(a^2-4*b)^(1/2))-1/2*(a^2-4*b)^(1/2)*(ln(b*x^2*y(x)^2
+a*x*y(x)+1)-2*_C1+2*ln(x)-2*ln(x*y(x))))/(a^2-4*b)^(1/2) = 0