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

Question

anonymous · Answer

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

anonymous · Answer

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

unklerhaukus · Answer

yes, you can combine the $dy$ terms,

sirm3d · Answer

$$ydx+xdy+x^2y^5dy=0\d(xy)+x^2y^5dy=0$$
divide by $x^2y^2$
$$\frac{d(xy)}{(xy)^2}+y^3dy=0$$

integrate both terms

anonymous · Answer

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?

sirm3d · Answer

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

sirm3d · Answer

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

anonymous · Answer

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