4.6.15 \(a x^2+b x y(x)+c y(x)^2+x^2 y'(x)=0\)

ODE
\[ a x^2+b x y(x)+c y(x)^2+x^2 y'(x)=0 \] ODE Classification

[[_homogeneous, `class A`], _rational, _Riccati]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0634027 (sec), leaf count = 66

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

Maple
cpu = 0.014 (sec), leaf count = 60

\[ \left \{ -2\,{\frac {1}{\sqrt {4\,ca-{b}^{2}-2\,b-1}}\arctan \left ({\frac {bx+2\,cy \relax (x ) +x}{x\sqrt {4\,ca-{b}^{2}-2\,b-1}}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[a*x^2 + b*x*y[x] + c*y[x]^2 + x^2*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(x*(1 + b - Sqrt[-1 - 2*b - b^2 + 4*a*c]*Tan[(Sqrt[-1 - 2*b - b^2 + 4
*a*c]*(C[1] - Log[x]))/2]))/(2*c)}}

Maple raw input

dsolve(x^2*diff(y(x),x)+a*x^2+b*x*y(x)+c*y(x)^2 = 0, y(x),'implicit')

Maple raw output

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