4.36.5 $y^{″}(x){(a+bx+c{x}^2)}^2+Ay(x)=0$

ODE
$$y^{″}(x){(a+bx+c{x}^2)}^2+Ay(x)=0$$ ODE Classification

[[_Emden, _Fowler]]

Book solution method
TO DO

Mathematica ✓
cpu = 1.46083 (sec), leaf count = 199

$$\left\{\{y(x)\to \sqrt{a+x(b+cx)}\exp (-\frac{\sqrt{4ac-b^2}\sqrt{1-\frac{4A}{b^2-4ac}}{\tan }^{-1}\left(\frac{b+2cx}{\sqrt{4ac-b^2}}\right)}{\sqrt{b^2-4ac}})(c_1\exp \left(\frac{2\sqrt{4ac-b^2}\sqrt{1-\frac{4A}{b^2-4ac}}{\tan }^{-1}\left(\frac{b+2cx}{\sqrt{4ac-b^2}}\right)}{\sqrt{b^2-4ac}}\right)+\frac{c_2}{\sqrt{b^2-4ac}\sqrt{1-\frac{4A}{b^2-4ac}}})\}\right\}$$

Maple ✓
cpu = 0.148 (sec), leaf count = 178

$$\{y\left(x\right)=\sqrt{c{x}^2+bx+a}({\left(1(i\sqrt{4ac-b^2}-2cx-b){(2cx+b+i\sqrt{4ac-b^2})}^{-1}\right)}^{-\frac{c}{2}\sqrt{\frac{-4ac+b^2-4A}{c^2}}\frac{1}{\sqrt{-4ac+b^2}}}\mathit{\_}\mathit{C}\mathit{2}+{\left(1(i\sqrt{4ac-b^2}-2cx-b){(2cx+b+i\sqrt{4ac-b^2})}^{-1}\right)}^{\frac{c}{2}\sqrt{\frac{-4ac+b^2-4A}{c^2}}\frac{1}{\sqrt{-4ac+b^2}}}\mathit{\_}\mathit{C}\mathit{1})\}$$ 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)