4.15.18 \(\left (x^2-y(x)^5\right ) y'(x)=2 x y(x)\)

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

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0127628 (sec), leaf count = 116

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

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

\[ \left \{ \ln \relax (x ) -{\it \_C1}-{\frac {5}{8}\ln \left ({y \relax (x ) {x}^{-{\frac {2}{5}}}} \right ) }+{\frac {5}{8}\ln \left ({\frac { \left (y \relax (x ) \right ) ^{5}+4\,{x}^{2}}{{x}^{2}}} \right ) }=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

ln(x)-_C1-5/8*ln(y(x)/x^(2/5))+5/8*ln((y(x)^5+4*x^2)/x^2) = 0