4.15.16 \(2 x \left (x^3+y(x)^4\right ) y'(x)=y(x) \left (x^3+2 y(x)^4\right )\)

ODE
\[ 2 x \left (x^3+y(x)^4\right ) y'(x)=y(x) \left (x^3+2 y(x)^4\right ) \] ODE Classification

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0246659 (sec), leaf count = 161

\[\left \{\left \{y(x)\to -\frac {\sqrt {c_1 x^2-x^{3/2} \sqrt {c_1^2 x+4}}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {c_1 x^2-x^{3/2} \sqrt {c_1^2 x+4}}}{\sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {x^{3/2} \sqrt {c_1^2 x+4}+c_1 x^2}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {x^{3/2} \sqrt {c_1^2 x+4}+c_1 x^2}}{\sqrt {2}}\right \}\right \}\]

Maple
cpu = 0.035 (sec), leaf count = 61

\[ \left \{ \ln \relax (x ) -{\it \_C1}-2\,\ln \left ({\frac {y \relax (x ) -{x}^{3/4}}{{x}^{3/4}}} \right ) -2\,\ln \left ({\frac {{x}^{3/4}+y \relax (x ) }{{x}^{3/4}}} \right ) -2\,\ln \left ({\frac {{x}^{3/2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{3/2}}} \right ) +4\,\ln \left ({\frac {y \relax (x ) }{{x}^{3/4}}} \right ) =0 \right \} \] Mathematica raw input

DSolve[2*x*(x^3 + y[x]^4)*y'[x] == y[x]*(x^3 + 2*y[x]^4),y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[x^2*C[1] - x^(3/2)*Sqrt[4 + x*C[1]^2]]/Sqrt[2])}, {y[x] -> Sqrt
[x^2*C[1] - x^(3/2)*Sqrt[4 + x*C[1]^2]]/Sqrt[2]}, {y[x] -> -(Sqrt[x^2*C[1] + x^(
3/2)*Sqrt[4 + x*C[1]^2]]/Sqrt[2])}, {y[x] -> Sqrt[x^2*C[1] + x^(3/2)*Sqrt[4 + x*
C[1]^2]]/Sqrt[2]}}

Maple raw input

dsolve(2*x*(x^3+y(x)^4)*diff(y(x),x) = (x^3+2*y(x)^4)*y(x), y(x),'implicit')

Maple raw output

ln(x)-_C1-2*ln((y(x)-x^(3/4))/x^(3/4))-2*ln((x^(3/4)+y(x))/x^(3/4))-2*ln((x^(3/2
)+y(x)^2)/x^(3/2))+4*ln(y(x)/x^(3/4)) = 0