4.13.42 \(x \left (3 x-y(x)^2\right ) y'(x)+y(x) \left (5 x-2 y(x)^2\right )=0\)

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

[[_homogeneous, `class G`], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0915318 (sec), leaf count = 661

\[\left \{\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,6\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,7\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,8\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,9\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,10\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,11\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,12\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,13\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,14\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {$\#$1}^{15}-\frac {25 \text {$\#$1}^2 e^{\frac {65 c_1}{2}}}{x^{26}}+\frac {65 e^{\frac {65 c_1}{2}}}{x^{25}}\& ,15\right ]\right \}\right \}\]

Maple
cpu = 0.02 (sec), leaf count = 35

\[ \left \{ \ln \relax (x ) -{\it \_C1}-{\frac {2}{65}\ln \left ({\frac {5\, \left (y \relax (x ) \right ) ^{2}-13\,x}{x}} \right ) }+{\frac {6}{13}\ln \left ({y \relax (x ) {\frac {1}{\sqrt {x}}}} \right ) }=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> Root[(65*E^((65*C[1])/2))/x^25 - (25*E^((65*C[1])/2)*#1^2)/x^26 - #1^1
5 & , 1]}, {y[x] -> Root[(65*E^((65*C[1])/2))/x^25 - (25*E^((65*C[1])/2)*#1^2)/x
^26 - #1^15 & , 2]}, {y[x] -> Root[(65*E^((65*C[1])/2))/x^25 - (25*E^((65*C[1])/
2)*#1^2)/x^26 - #1^15 & , 3]}, {y[x] -> Root[(65*E^((65*C[1])/2))/x^25 - (25*E^(
(65*C[1])/2)*#1^2)/x^26 - #1^15 & , 4]}, {y[x] -> Root[(65*E^((65*C[1])/2))/x^25
 - (25*E^((65*C[1])/2)*#1^2)/x^26 - #1^15 & , 5]}, {y[x] -> Root[(65*E^((65*C[1]
)/2))/x^25 - (25*E^((65*C[1])/2)*#1^2)/x^26 - #1^15 & , 6]}, {y[x] -> Root[(65*E
^((65*C[1])/2))/x^25 - (25*E^((65*C[1])/2)*#1^2)/x^26 - #1^15 & , 7]}, {y[x] -> 
Root[(65*E^((65*C[1])/2))/x^25 - (25*E^((65*C[1])/2)*#1^2)/x^26 - #1^15 & , 8]},
 {y[x] -> Root[(65*E^((65*C[1])/2))/x^25 - (25*E^((65*C[1])/2)*#1^2)/x^26 - #1^1
5 & , 9]}, {y[x] -> Root[(65*E^((65*C[1])/2))/x^25 - (25*E^((65*C[1])/2)*#1^2)/x
^26 - #1^15 & , 10]}, {y[x] -> Root[(65*E^((65*C[1])/2))/x^25 - (25*E^((65*C[1])
/2)*#1^2)/x^26 - #1^15 & , 11]}, {y[x] -> Root[(65*E^((65*C[1])/2))/x^25 - (25*E
^((65*C[1])/2)*#1^2)/x^26 - #1^15 & , 12]}, {y[x] -> Root[(65*E^((65*C[1])/2))/x
^25 - (25*E^((65*C[1])/2)*#1^2)/x^26 - #1^15 & , 13]}, {y[x] -> Root[(65*E^((65*
C[1])/2))/x^25 - (25*E^((65*C[1])/2)*#1^2)/x^26 - #1^15 & , 14]}, {y[x] -> Root[
(65*E^((65*C[1])/2))/x^25 - (25*E^((65*C[1])/2)*#1^2)/x^26 - #1^15 & , 15]}}

Maple raw input

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

Maple raw output

ln(x)-_C1-2/65*ln((5*y(x)^2-13*x)/x)+6/13*ln(y(x)/x^(1/2)) = 0