ODE
\[ y''(x) \left (a+b x+c x^2\right )^2+A y(x)=0 \] ODE Classification
[[_Emden, _Fowler]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.35447 (sec), leaf count = 199
\[\left \{\left \{y(x)\to \sqrt {a+x (b+c x)} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right ) \left (c_1 \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+\frac {c_2}{\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.466 (sec), leaf count = 189
\[\left [y \left (x \right ) = \textit {\_C1} \sqrt {c \,x^{2}+b x +a}\, \left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )^{\frac {c \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+\textit {\_C2} \sqrt {c \,x^{2}+b x +a}\, \left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )^{-\frac {c \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right ]\] Mathematica raw input
DSolve[A*y[x] + (a + b*x + c*x^2)^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (Sqrt[a + x*(b + c*x)]*(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] + C[2]/(Sqrt[b^2 - 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)])))/E^((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])}}
Maple raw input
dsolve((c*x^2+b*x+a)^2*diff(diff(y(x),x),x)+A*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(c*x^2+b*x+a)^(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))+_C2*(c*x^2+b*x+a)^(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))]