##### 4.14.14 $$x \left (3 x-7 y(x)^2\right ) y'(x)+y(x) \left (5 x-3 y(x)^2\right )=0$$

ODE
$x \left (3 x-7 y(x)^2\right ) y'(x)+y(x) \left (5 x-3 y(x)^2\right )=0$ ODE Classiﬁcation

[[_homogeneous, class G], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.353801 (sec), leaf count = 288

$\left \{\left \{y(x)\to \text {Root}\left [4 \text {\#1}^7 x^3-8 \text {\#1}^5 x^4+4 \text {\#1}^3 x^5-c_1{}^2\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {\#1}^7 x^3-8 \text {\#1}^5 x^4+4 \text {\#1}^3 x^5-c_1{}^2\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {\#1}^7 x^3-8 \text {\#1}^5 x^4+4 \text {\#1}^3 x^5-c_1{}^2\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {\#1}^7 x^3-8 \text {\#1}^5 x^4+4 \text {\#1}^3 x^5-c_1{}^2\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {\#1}^7 x^3-8 \text {\#1}^5 x^4+4 \text {\#1}^3 x^5-c_1{}^2\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {\#1}^7 x^3-8 \text {\#1}^5 x^4+4 \text {\#1}^3 x^5-c_1{}^2\& ,6\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {\#1}^7 x^3-8 \text {\#1}^5 x^4+4 \text {\#1}^3 x^5-c_1{}^2\& ,7\right ]\right \}\right \}$

Maple
cpu = 0.509 (sec), leaf count = 49

$\left [y \left (x \right ) = \RootOf \left (x^{\frac {3}{2}} \textit {\_Z}^{7}-x^{\frac {5}{2}} \textit {\_Z}^{3}-\textit {\_C1} \right )^{2}, y \left (x \right ) = \RootOf \left (x^{\frac {3}{2}} \textit {\_Z}^{7}-x^{\frac {5}{2}} \textit {\_Z}^{3}+\textit {\_C1} \right )^{2}\right ]$ Mathematica raw input

DSolve[y[x]*(5*x - 3*y[x]^2) + x*(3*x - 7*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> Root[-C[1]^2 + 4*x^5*#1^3 - 8*x^4*#1^5 + 4*x^3*#1^7 & , 1]}, {y[x] ->
Root[-C[1]^2 + 4*x^5*#1^3 - 8*x^4*#1^5 + 4*x^3*#1^7 & , 2]}, {y[x] -> Root[-C[1]
^2 + 4*x^5*#1^3 - 8*x^4*#1^5 + 4*x^3*#1^7 & , 3]}, {y[x] -> Root[-C[1]^2 + 4*x^5
*#1^3 - 8*x^4*#1^5 + 4*x^3*#1^7 & , 4]}, {y[x] -> Root[-C[1]^2 + 4*x^5*#1^3 - 8*
x^4*#1^5 + 4*x^3*#1^7 & , 5]}, {y[x] -> Root[-C[1]^2 + 4*x^5*#1^3 - 8*x^4*#1^5 +
 4*x^3*#1^7 & , 6]}, {y[x] -> Root[-C[1]^2 + 4*x^5*#1^3 - 8*x^4*#1^5 + 4*x^3*#1^
7 & , 7]}}

Maple raw input

dsolve(x*(3*x-7*y(x)^2)*diff(y(x),x)+(5*x-3*y(x)^2)*y(x) = 0, y(x))

Maple raw output

[y(x) = RootOf(x^(3/2)*_Z^7-x^(5/2)*_Z^3-_C1)^2, y(x) = RootOf(x^(3/2)*_Z^7-x^(5
/2)*_Z^3+_C1)^2]