4.8.30 \(\left (y'(x)+y(x)^2\right ) \left (a+b x+c x^2\right )^2+A=0\)

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

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 1.46923 (sec), leaf count = 612

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

Maple
cpu = 0.383 (sec), leaf count = 490

\[ \left \{ y \relax (x ) =2\,{\frac {c}{\sqrt {-4\,ca+{b}^{2}} \left (2\,cx+b+i\sqrt {4\,ca-{b}^{2}} \right ) \left (i\sqrt {4\,ca-{b}^{2}}-2\,cx-b \right ) } \left (\left (i\sqrt {{\frac {-4\,ca+{b}^{2}-4\,A}{{c}^{2}}}}c\sqrt {4\,ca-{b}^{2}}-\sqrt {-4\,ca+{b}^{2}} \left (2\,cx+b \right ) \right ) {\it \_C1}\, \left ({\frac {i\sqrt {4\,ca-{b}^{2}}-2\,cx-b}{2\,cx+b+i\sqrt {4\,ca-{b}^{2}}}} \right ) ^{-1/2\,{\frac {c}{\sqrt {-4\,ca+{b}^{2}}}\sqrt {{\frac {-4\,ca+{b}^{2}-4\,A}{{c}^{2}}}}}}- \left ({\frac {i\sqrt {4\,ca-{b}^{2}}-2\,cx-b}{2\,cx+b+i\sqrt {4\,ca-{b}^{2}}}} \right ) ^{1/2\,{\frac {c}{\sqrt {-4\,ca+{b}^{2}}}\sqrt {{\frac {-4\,ca+{b}^{2}-4\,A}{{c}^{2}}}}}} \left (i\sqrt {{\frac {-4\,ca+{b}^{2}-4\,A}{{c}^{2}}}}c\sqrt {4\,ca-{b}^{2}}+\sqrt {-4\,ca+{b}^{2}} \left (2\,cx+b \right ) \right ) \right ) \left ({\it \_C1}\, \left ({\frac {i\sqrt {4\,ca-{b}^{2}}-2\,cx-b}{2\,cx+b+i\sqrt {4\,ca-{b}^{2}}}} \right ) ^{-1/2\,{\frac {c}{\sqrt {-4\,ca+{b}^{2}}}\sqrt {{\frac {-4\,ca+{b}^{2}-4\,A}{{c}^{2}}}}}}+ \left ({\frac {i\sqrt {4\,ca-{b}^{2}}-2\,cx-b}{2\,cx+b+i\sqrt {4\,ca-{b}^{2}}}} \right ) ^{1/2\,{\frac {c}{\sqrt {-4\,ca+{b}^{2}}}\sqrt {{\frac {-4\,ca+{b}^{2}-4\,A}{{c}^{2}}}}}} \right ) ^{-1}} \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (b + Sqrt[b^2 - 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)] + 2*c*x + 4*A*E^(
(2*Sqrt[-b^2 + 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*ArcTan[(b + 2*c*x)/Sqrt[-b^2
 + 4*a*c]])/Sqrt[b^2 - 4*a*c])*C[1] - b^2*E^((2*Sqrt[-b^2 + 4*a*c]*Sqrt[1 - (4*A
)/(b^2 - 4*a*c)]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[b^2 - 4*a*c])*C[1]
 + 4*a*c*E^((2*Sqrt[-b^2 + 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*ArcTan[(b + 2*c*
x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[b^2 - 4*a*c])*C[1] + b*Sqrt[b^2 - 4*a*c]*Sqrt[1 - (
4*A)/(b^2 - 4*a*c)]*E^((2*Sqrt[-b^2 + 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*ArcTa
n[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[b^2 - 4*a*c])*C[1] + 2*c*Sqrt[b^2 - 4*a*
c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*E^((2*Sqrt[-b^2 + 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 
4*a*c)]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[b^2 - 4*a*c])*x*C[1])/(2*(a
 + x*(b + c*x))*(1 + Sqrt[b^2 - 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*E^((2*Sqrt[
-b^2 + 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c
]])/Sqrt[b^2 - 4*a*c])*C[1]))}}

Maple raw input

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

Maple raw output

y(x) = 2*((I*((-4*a*c+b^2-4*A)/c^2)^(1/2)*c*(4*a*c-b^2)^(1/2)-(-4*a*c+b^2)^(1/2)
*(2*c*x+b))*_C1*((I*(4*a*c-b^2)^(1/2)-2*c*x-b)/(2*c*x+b+I*(4*a*c-b^2)^(1/2)))^(-
1/2*c/(-4*a*c+b^2)^(1/2)*((-4*a*c+b^2-4*A)/c^2)^(1/2))-((I*(4*a*c-b^2)^(1/2)-2*c
*x-b)/(2*c*x+b+I*(4*a*c-b^2)^(1/2)))^(1/2*c/(-4*a*c+b^2)^(1/2)*((-4*a*c+b^2-4*A)
/c^2)^(1/2))*(I*((-4*a*c+b^2-4*A)/c^2)^(1/2)*c*(4*a*c-b^2)^(1/2)+(-4*a*c+b^2)^(1
/2)*(2*c*x+b)))*c/(-4*a*c+b^2)^(1/2)/(2*c*x+b+I*(4*a*c-b^2)^(1/2))/(I*(4*a*c-b^2
)^(1/2)-2*c*x-b)/(_C1*((I*(4*a*c-b^2)^(1/2)-2*c*x-b)/(2*c*x+b+I*(4*a*c-b^2)^(1/2
)))^(-1/2*c/(-4*a*c+b^2)^(1/2)*((-4*a*c+b^2-4*A)/c^2)^(1/2))+((I*(4*a*c-b^2)^(1/
2)-2*c*x-b)/(2*c*x+b+I*(4*a*c-b^2)^(1/2)))^(1/2*c/(-4*a*c+b^2)^(1/2)*((-4*a*c+b^
2-4*A)/c^2)^(1/2)))