Question

2x^3y dx + (x^4 + y^4) dy =0 since the coefficient of dx is slightly simpler than coffeicient of dy, we try x=vy after substituting

Answer 1

is it 2x^3y

Answer 2

put x=vy
\[\frac{ dx }{ dy }=v+y \frac{ dv }{ dy }\]
\[v+\frac{ dv }{ dy }=\frac{ v^4 y^4+y^4 }{ 2 v^3 y^4 }=\frac{ v^4+1 }{ 2 v^3 }\]
\[\frac{ dv }{ dy }=\frac{ v^4+1-2 v^4 }{ 2 v^3 }=\frac{ 1-v^4 }{ 2 v^3 }\]
separating the variables and integrating

Answer 3

\[\int\limits \frac{ 2 v^3 }{ 1-v^4 }dv=\int\limits dy+c\]

Answer 4

\[-\frac{ 1 }{ 2 }\int\limits \frac{ -4v^3 ~dv }{ 1-v^4 }=y+c\]
\[-\frac{ 1 }{ 2 }\ln \left| 1-v^4 \right|=y+c\]
replace the value of v