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 = 0.0485907 (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 = 0.028 (sec), leaf count = 46

\[ \left \{ \ln \relax (x ) -{\it \_C1}+{\frac {4}{13}\ln \left ({1 \left (y \relax (x ) -\sqrt {x} \right ) {\frac {1}{\sqrt {x}}}} \right ) }+{\frac {4}{13}\ln \left ({1 \left (y \relax (x ) +\sqrt {x} \right ) {\frac {1}{\sqrt {x}}}} \right ) }+{\frac {6}{13}\ln \left ({y \relax (x ) {\frac {1}{\sqrt {x}}}} \right ) }=0 \right \} \] 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),'implicit')

Maple raw output

ln(x)-_C1+4/13*ln((y(x)-x^(1/2))/x^(1/2))+4/13*ln((y(x)+x^(1/2))/x^(1/2))+6/13*l
n(y(x)/x^(1/2)) = 0