##### 4.17.7 $$a x^2 y'(x)+b x y(x)+y'(x)^2=0$$

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

[[_homogeneous, class G]]

Book solution method
Homogeneous ODE, The Isobaric equation

Mathematica
cpu = 2.80997 (sec), leaf count = 559

$\left \{\text {Solve}\left [c_1=\int _1^{y(x)}\frac {a x^3-\sqrt {a^2 x^4-4 b x K[2]} x+6 K[2]-2 K[2] \left (3 a x^3+b x^3+9 K[2]\right ) \int _1^x-\frac {3 K[1] \left (9 a^2 K[1]^3+2 b^2 K[1]^3+6 a b K[1]^3+9 a \sqrt {a^2 K[1]^4-4 b K[1] K[2]} K[1]+6 b \sqrt {a^2 K[1]^4-4 b K[1] K[2]} K[1]-18 b K[2]\right )}{2 \left (3 a K[1]^3+b K[1]^3+9 K[2]\right )^2 \sqrt {a^2 K[1]^4-4 b K[1] K[2]}}dK[1]}{2 K[2] \left (3 a x^3+b x^3+9 K[2]\right )}dK[2]+\int _1^x\frac {3 a K[1]^2+2 b K[1]^2+3 \sqrt {a^2 K[1]^4-4 b K[1] y(x)}}{2 \left ((3 a+b) K[1]^3+9 y(x)\right )}dK[1],y(x)\right ],\text {Solve}\left [c_1=\int _1^{y(x)}\frac {a x^3+\sqrt {a^2 x^4-4 b x K[4]} x+6 K[4]-2 K[4] \left (3 a x^3+b x^3+9 K[4]\right ) \int _1^x\frac {3 \left (-3 (3 a+2 b) K[3]^2+\frac {2 b \left (3 a K[3]^3+b K[3]^3+9 K[4]\right ) K[3]}{\sqrt {a^2 K[3]^4-4 b K[3] K[4]}}+9 \sqrt {a^2 K[3]^4-4 b K[3] K[4]}\right )}{2 \left (3 a K[3]^3+b K[3]^3+9 K[4]\right )^2}dK[3]}{2 K[4] \left (3 a x^3+b x^3+9 K[4]\right )}dK[4]+\int _1^x\frac {3 a K[3]^2+2 b K[3]^2-3 \sqrt {a^2 K[3]^4-4 b K[3] y(x)}}{2 \left ((3 a+b) K[3]^3+9 y(x)\right )}dK[3],y(x)\right ]\right \}$

Maple
cpu = 0.487 (sec), leaf count = 465

$\left [\int _{\textit {\_b}}^{x}-\frac {a \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} b y \left (x \right )}}{a \,\textit {\_a}^{3}+\textit {\_a} \sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} b y \left (x \right )}+6 y \left (x \right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (-\frac {2}{a \,x^{3}+x \sqrt {a^{2} x^{4}-4 \textit {\_f} b x}+6 \textit {\_f}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {2 b \textit {\_a}}{\sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} \textit {\_f} b}\, \left (a \,\textit {\_a}^{3}+\textit {\_a} \sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} \textit {\_f} b}+6 \textit {\_f} \right )}+\frac {\left (a \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} \textit {\_f} b}\right ) \left (-\frac {2 \textit {\_a}^{2} b}{\sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} \textit {\_f} b}}+6\right )}{\left (a \,\textit {\_a}^{3}+\textit {\_a} \sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} \textit {\_f} b}+6 \textit {\_f} \right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +\textit {\_C1} = 0, \int _{\textit {\_b}}^{x}-\frac {-a \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} b y \left (x \right )}}{-a \,\textit {\_a}^{3}+\textit {\_a} \sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} b y \left (x \right )}-6 y \left (x \right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (\frac {2}{-a \,x^{3}+x \sqrt {a^{2} x^{4}-4 \textit {\_f} b x}-6 \textit {\_f}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {2 b \textit {\_a}}{\sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} \textit {\_f} b}\, \left (-a \,\textit {\_a}^{3}+\textit {\_a} \sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} \textit {\_f} b}-6 \textit {\_f} \right )}+\frac {\left (-a \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} \textit {\_f} b}\right ) \left (-\frac {2 \textit {\_a}^{2} b}{\sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} \textit {\_f} b}}-6\right )}{\left (-a \,\textit {\_a}^{3}+\textit {\_a} \sqrt {\textit {\_a}^{4} a^{2}-4 \textit {\_a} \textit {\_f} b}-6 \textit {\_f} \right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{Solve[C[1] == Inactive[Integrate][(3*a*K[1]^2 + 2*b*K[1]^2 + 3*Sqrt[a^2*K[1]^4
- 4*b*K[1]*y[x]])/(2*((3*a + b)*K[1]^3 + 9*y[x])), {K[1], 1, x}] + Inactive[Inte
grate][(a*x^3 + 6*K[2] - x*Sqrt[a^2*x^4 - 4*b*x*K[2]] - 2*K[2]*(3*a*x^3 + b*x^3
+ 9*K[2])*Inactive[Integrate][(-3*K[1]*(9*a^2*K[1]^3 + 6*a*b*K[1]^3 + 2*b^2*K[1]
^3 - 18*b*K[2] + 9*a*K[1]*Sqrt[a^2*K[1]^4 - 4*b*K[1]*K[2]] + 6*b*K[1]*Sqrt[a^2*K
[1]^4 - 4*b*K[1]*K[2]]))/(2*(3*a*K[1]^3 + b*K[1]^3 + 9*K[2])^2*Sqrt[a^2*K[1]^4 -
 4*b*K[1]*K[2]]), {K[1], 1, x}])/(2*K[2]*(3*a*x^3 + b*x^3 + 9*K[2])), {K[2], 1,
y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(3*a*K[3]^2 + 2*b*K[3]^2 - 3*Sq
rt[a^2*K[3]^4 - 4*b*K[3]*y[x]])/(2*((3*a + b)*K[3]^3 + 9*y[x])), {K[3], 1, x}] +
 Inactive[Integrate][(a*x^3 + 6*K[4] + x*Sqrt[a^2*x^4 - 4*b*x*K[4]] - 2*K[4]*(3*
a*x^3 + b*x^3 + 9*K[4])*Inactive[Integrate][(3*(-3*(3*a + 2*b)*K[3]^2 + (2*b*K[3
]*(3*a*K[3]^3 + b*K[3]^3 + 9*K[4]))/Sqrt[a^2*K[3]^4 - 4*b*K[3]*K[4]] + 9*Sqrt[a^
2*K[3]^4 - 4*b*K[3]*K[4]]))/(2*(3*a*K[3]^3 + b*K[3]^3 + 9*K[4])^2), {K[3], 1, x}
])/(2*K[4]*(3*a*x^3 + b*x^3 + 9*K[4])), {K[4], 1, y[x]}], y[x]]}

Maple raw input

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

Maple raw output

[Int(-(a*_a^2+(_a^4*a^2-4*_a*b*y(x))^(1/2))/(a*_a^3+_a*(_a^4*a^2-4*_a*b*y(x))^(1
/2)+6*y(x)),_a = _b .. x)+Intat(-2/(a*x^3+x*(a^2*x^4-4*_f*b*x)^(1/2)+6*_f)-Int(2
/(_a^4*a^2-4*_a*_f*b)^(1/2)*b*_a/(a*_a^3+_a*(_a^4*a^2-4*_a*_f*b)^(1/2)+6*_f)+(a*
_a^2+(_a^4*a^2-4*_a*_f*b)^(1/2))/(a*_a^3+_a*(_a^4*a^2-4*_a*_f*b)^(1/2)+6*_f)^2*(
-2*_a^2/(_a^4*a^2-4*_a*_f*b)^(1/2)*b+6),_a = _b .. x),_f = y(x))+_C1 = 0, Int(-(
-a*_a^2+(_a^4*a^2-4*_a*b*y(x))^(1/2))/(-a*_a^3+_a*(_a^4*a^2-4*_a*b*y(x))^(1/2)-6
*y(x)),_a = _b .. x)+Intat(2/(-a*x^3+x*(a^2*x^4-4*_f*b*x)^(1/2)-6*_f)-Int(2/(_a^
4*a^2-4*_a*_f*b)^(1/2)*b*_a/(-a*_a^3+_a*(_a^4*a^2-4*_a*_f*b)^(1/2)-6*_f)+(-a*_a^
2+(_a^4*a^2-4*_a*_f*b)^(1/2))/(-a*_a^3+_a*(_a^4*a^2-4*_a*_f*b)^(1/2)-6*_f)^2*(-2
*_a^2/(_a^4*a^2-4*_a*_f*b)^(1/2)*b-6),_a = _b .. x),_f = y(x))+_C1 = 0]