##### 4.17.4 $$a x y'(x)+b x^2+c y(x)+y'(x)^2=0$$

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

[[_homogeneous, class G]]

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

Mathematica
cpu = 6.26663 (sec), leaf count = 779

$\left \{\text {Solve}\left [c_1=\int _1^{y(x)}\frac {a x^2-\sqrt {a^2 x^2-4 b x^2-4 c K[2]} x+4 K[2]-2 \left (b x^4+K[2] \left (2 a x^2+c x^2+4 K[2]\right )\right ) \int _1^x\frac {\left (b K[1]^6-4 K[1]^2 K[2]^2\right ) a^2-K[1] \left (4 c K[1] K[2]^2+\sqrt {a^2 K[1]^2-4 b K[1]^2-4 c K[2]} \left (b K[1]^4+4 K[2]^2\right )\right ) a-2 \left (2 b K[1]^2+c K[2]\right ) \left (b K[1]^4+K[2] \left (c K[1]^2+2 \sqrt {a^2 K[1]^2-4 b K[1]^2-4 c K[2]} K[1]-4 K[2]\right )\right )}{\sqrt {a^2 K[1]^2-4 b K[1]^2-4 c K[2]} \left (b K[1]^4+K[2] \left (2 a K[1]^2+c K[1]^2+4 K[2]\right )\right )^2}dK[1]}{2 \left (b x^4+K[2] \left (2 a x^2+c x^2+4 K[2]\right )\right )}dK[2]+\int _1^x\frac {b K[1]^3+y(x) \left (a K[1]+c K[1]+\sqrt {\left (a^2-4 b\right ) K[1]^2-4 c y(x)}\right )}{b K[1]^4+(2 a+c) y(x) K[1]^2+4 y(x)^2}dK[1],y(x)\right ],\text {Solve}\left [c_1=\int _1^{y(x)}\frac {a x^2+\sqrt {a^2 x^2-4 b x^2-4 c K[4]} x+4 K[4]-2 \left (b x^4+K[4] \left (2 a x^2+c x^2+4 K[4]\right )\right ) \int _1^x\frac {\left (4 K[3]^2 K[4]^2-b K[3]^6\right ) a^2-K[3] \left (\sqrt {a^2 K[3]^2-4 b K[3]^2-4 c K[4]} \left (b K[3]^4+4 K[4]^2\right )-4 c K[3] K[4]^2\right ) a+2 \left (2 b K[3]^2+c K[4]\right ) \left (b K[3]^4+K[4] \left (c K[3]^2-2 \sqrt {a^2 K[3]^2-4 b K[3]^2-4 c K[4]} K[3]-4 K[4]\right )\right )}{\sqrt {a^2 K[3]^2-4 b K[3]^2-4 c K[4]} \left (b K[3]^4+K[4] \left (2 a K[3]^2+c K[3]^2+4 K[4]\right )\right )^2}dK[3]}{2 \left (b x^4+K[4] \left (2 a x^2+c x^2+4 K[4]\right )\right )}dK[4]+\int _1^x\frac {b K[3]^3+y(x) \left (a K[3]+c K[3]-\sqrt {\left (a^2-4 b\right ) K[3]^2-4 c y(x)}\right )}{b K[3]^4+(2 a+c) y(x) K[3]^2+4 y(x)^2}dK[3],y(x)\right ]\right \}$

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

time expired

Mathematica raw input

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

Mathematica raw output

{Solve[C[1] == Inactive[Integrate][(b*K[1]^3 + y[x]*(a*K[1] + c*K[1] + Sqrt[(a^2
 - 4*b)*K[1]^2 - 4*c*y[x]]))/(b*K[1]^4 + (2*a + c)*K[1]^2*y[x] + 4*y[x]^2), {K[1
], 1, x}] + Inactive[Integrate][(a*x^2 + 4*K[2] - x*Sqrt[a^2*x^2 - 4*b*x^2 - 4*c
*K[2]] - 2*(b*x^4 + K[2]*(2*a*x^2 + c*x^2 + 4*K[2]))*Inactive[Integrate][(a^2*(b
*K[1]^6 - 4*K[1]^2*K[2]^2) - a*K[1]*(4*c*K[1]*K[2]^2 + Sqrt[a^2*K[1]^2 - 4*b*K[1
]^2 - 4*c*K[2]]*(b*K[1]^4 + 4*K[2]^2)) - 2*(2*b*K[1]^2 + c*K[2])*(b*K[1]^4 + K[2
]*(c*K[1]^2 - 4*K[2] + 2*K[1]*Sqrt[a^2*K[1]^2 - 4*b*K[1]^2 - 4*c*K[2]])))/(Sqrt[
a^2*K[1]^2 - 4*b*K[1]^2 - 4*c*K[2]]*(b*K[1]^4 + K[2]*(2*a*K[1]^2 + c*K[1]^2 + 4*
K[2]))^2), {K[1], 1, x}])/(2*(b*x^4 + K[2]*(2*a*x^2 + c*x^2 + 4*K[2]))), {K[2],
1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(b*K[3]^3 + y[x]*(a*K[3] + c
*K[3] - Sqrt[(a^2 - 4*b)*K[3]^2 - 4*c*y[x]]))/(b*K[3]^4 + (2*a + c)*K[3]^2*y[x]
+ 4*y[x]^2), {K[3], 1, x}] + Inactive[Integrate][(a*x^2 + 4*K[4] + x*Sqrt[a^2*x^
2 - 4*b*x^2 - 4*c*K[4]] - 2*(b*x^4 + K[4]*(2*a*x^2 + c*x^2 + 4*K[4]))*Inactive[I
ntegrate][(a^2*(-(b*K[3]^6) + 4*K[3]^2*K[4]^2) - a*K[3]*(-4*c*K[3]*K[4]^2 + Sqrt
[a^2*K[3]^2 - 4*b*K[3]^2 - 4*c*K[4]]*(b*K[3]^4 + 4*K[4]^2)) + 2*(2*b*K[3]^2 + c*
K[4])*(b*K[3]^4 + K[4]*(c*K[3]^2 - 4*K[4] - 2*K[3]*Sqrt[a^2*K[3]^2 - 4*b*K[3]^2
- 4*c*K[4]])))/(Sqrt[a^2*K[3]^2 - 4*b*K[3]^2 - 4*c*K[4]]*(b*K[3]^4 + K[4]*(2*a*K
[3]^2 + c*K[3]^2 + 4*K[4]))^2), {K[3], 1, x}])/(2*(b*x^4 + K[4]*(2*a*x^2 + c*x^2
 + 4*K[4]))), {K[4], 1, y[x]}], y[x]]}

Maple raw input

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

Maple raw output

\verbtime expired||