##### 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 = 0.442726 (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 = 0.377 (sec), leaf count = 36

$\left [\ln \left (x \right )-\textit {\_C1} +\frac {6 \ln \left (\frac {y \left (x \right )}{\sqrt {x}}\right )}{13}-\frac {2 \ln \left (-\frac {-5 y \left (x \right )^{2}+13 x}{x}\right )}{65} = 0\right ]$ 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))

Maple raw output

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