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 = 1.46083 (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 = 0.148 (sec), leaf count = 178
\[ \left \{ y \left ( x \right ) =\sqrt {c{x}^{2}+bx+a} \left ( \left ( {1 \left ( i\sqrt {4\,ac-{b}^{2}}-2\,cx-b \right ) \left ( 2\,cx+b+i\sqrt {4\,ac-{b}^{2}} \right ) ^{-1}} \right ) ^{-{\frac {c}{2}\sqrt {{\frac {-4\,ac+{b}^{2}-4\,A}{{c}^{2}}}}{\frac {1}{\sqrt {-4\,ac+{b}^{2}}}}}}{\it \_C2}+ \left ( {1 \left ( i\sqrt {4\,ac-{b}^{2}}-2\,cx-b \right ) \left ( 2\,cx+b+i\sqrt {4\,ac-{b}^{2}} \right ) ^{-1}} \right ) ^{{\frac {c}{2}\sqrt {{\frac {-4\,ac+{b}^{2}-4\,A}{{c}^{2}}}}{\frac {1}{\sqrt {-4\,ac+{b}^{2}}}}}}{\it \_C1} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = (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+((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))*_C1)