Question

Find all solutions in implicit form of the following ODE in symmetric form: \[x \ dy +y \ dx+ x ^{2}y ^{5} \ dy=0\]

Answer 1

Also am I able to combine x dy and x^2 y^5 dy?

Answer 2

@UnkleRhaukus @amistre64 @phi @Zarkon @experimentX @TuringTest

Answer 3

yes, you can combine the \(dy\) terms,

Answer 4

\[ydx+xdy+x^2y^5dy=0\\d(xy)+x^2y^5dy=0\]
divide by \(x^2y^2\)
\[\frac{d(xy)}{(xy)^2}+y^3dy=0\]

integrate both terms

Answer 5

That means I could rearrange it:
\[y ^{4} y'+ \frac{y ^{5}}{x} = - \frac{1}{x ^{2}}\]and solve it as a Bernoulli equation right?

Answer 6

\[-\frac{1}{xy}+\frac{y^4}{4}=C\]

Answer 7

if you put \[u=xy\] of the first term, it would look like \[\frac{du}{u^2}\]

Answer 8

Don't you have to check whether or not it's exact