##### 4.16.8 $$a x^2+b y(x)+y'(x)^2=0$$

ODE
$a x^2+b y(x)+y'(x)^2=0$ ODE Classiﬁcation

[[_homogeneous, class G]]

Book solution method
No Missing Variables ODE, Solve for $$y$$

Mathematica
cpu = 1.6311 (sec), leaf count = 498

$\left \{\text {Solve}\left [c_1=\int _1^{y(x)}-\frac {\sqrt {-a x^2-b K[2]} x-2 K[2]+\left (a x^4+b K[2] x^2+4 K[2]^2\right ) \int _1^x-\frac {\left (2 a K[1]^2+b K[2]\right ) \left (a K[1]^4+K[2] \left (b K[1]^2+4 \sqrt {-a K[1]^2-b K[2]} K[1]-4 K[2]\right )\right )}{\sqrt {-a K[1]^2-b K[2]} \left (a K[1]^4+b K[2] K[1]^2+4 K[2]^2\right )^2}dK[1]}{a x^4+b K[2] x^2+4 K[2]^2}dK[2]+\int _1^x\frac {a K[1]^3+b y(x) K[1]+2 y(x) \sqrt {-a K[1]^2-b y(x)}}{a K[1]^4+b y(x) K[1]^2+4 y(x)^2}dK[1],y(x)\right ],\text {Solve}\left [c_1=\int _1^{y(x)}\frac {\sqrt {-a x^2-b K[4]} x+2 K[4]-\left (a x^4+b K[4] x^2+4 K[4]^2\right ) \int _1^x\frac {\left (2 a K[3]^2+b K[4]\right ) \left (a K[3]^4+K[4] \left (b K[3]^2-4 \left (\sqrt {-a K[3]^2-b K[4]} K[3]+K[4]\right )\right )\right )}{\sqrt {-a K[3]^2-b K[4]} \left (a K[3]^4+b K[4] K[3]^2+4 K[4]^2\right )^2}dK[3]}{a x^4+b K[4] x^2+4 K[4]^2}dK[4]+\int _1^x\frac {a K[3]^3+b y(x) K[3]-2 y(x) \sqrt {-a K[3]^2-b y(x)}}{a K[3]^4+b y(x) K[3]^2+4 y(x)^2}dK[3],y(x)\right ]\right \}$

Maple
cpu = 0. (sec), leaf count = 0 , hanged

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

Mathematica raw input

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

Mathematica raw output

{Solve[C[1] == Inactive[Integrate][(a*K[1]^3 + b*K[1]*y[x] + 2*y[x]*Sqrt[-(a*K[1
]^2) - b*y[x]])/(a*K[1]^4 + b*K[1]^2*y[x] + 4*y[x]^2), {K[1], 1, x}] + Inactive[
Integrate][-((-2*K[2] + x*Sqrt[-(a*x^2) - b*K[2]] + (a*x^4 + b*x^2*K[2] + 4*K[2]
^2)*Inactive[Integrate][-(((2*a*K[1]^2 + b*K[2])*(a*K[1]^4 + K[2]*(b*K[1]^2 - 4*
K[2] + 4*K[1]*Sqrt[-(a*K[1]^2) - b*K[2]])))/(Sqrt[-(a*K[1]^2) - b*K[2]]*(a*K[1]^
4 + b*K[1]^2*K[2] + 4*K[2]^2)^2)), {K[1], 1, x}])/(a*x^4 + b*x^2*K[2] + 4*K[2]^2
)), {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(a*K[3]^3 + b*K[3
]*y[x] - 2*y[x]*Sqrt[-(a*K[3]^2) - b*y[x]])/(a*K[3]^4 + b*K[3]^2*y[x] + 4*y[x]^2
), {K[3], 1, x}] + Inactive[Integrate][(2*K[4] + x*Sqrt[-(a*x^2) - b*K[4]] - (a*
x^4 + b*x^2*K[4] + 4*K[4]^2)*Inactive[Integrate][((2*a*K[3]^2 + b*K[4])*(a*K[3]^
4 + K[4]*(b*K[3]^2 - 4*(K[4] + K[3]*Sqrt[-(a*K[3]^2) - b*K[4]]))))/(Sqrt[-(a*K[3
]^2) - b*K[4]]*(a*K[3]^4 + b*K[3]^2*K[4] + 4*K[4]^2)^2), {K[3], 1, x}])/(a*x^4 +
 b*x^2*K[4] + 4*K[4]^2), {K[4], 1, y[x]}], y[x]]}

Maple raw input

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

Maple raw output

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