SOLUTION: The differential equation y+y3=(y5+2x)y′ can be written in differential form: M(x,y)dx+N(x,y)dy=0 where M(x,y)= , and N(x,y)= . The term M(x,y)dx+N(x,y)dy bec

Let's write it this way





He tells us to divide through by y3 to make it an exact differential
equation.  So Let's multiply through by y-3 (same thing!)



Get the sign between them +



So

 and 

Now let's show that the DE is exact by showing the partial derivatives



,   

They are equal, so the differential equation is now exact.

So let's integrate both sides with respect to both variables.

Notice these 3 facts: 

1. When we integrate something containing dx with respect to the other letter y,
we get some function of the letter of its differential dx, the letter x, say u(x).
2. When we integrate something containing dy with respect to the other letter x,
we get some function of the letter of its differential dy, the letter y, say v(y).
3. When we integrate 0 with respect to either letter, we get the same constant C.

Let's integrate with respect to x:



Let's integrate with respect to y:



We must have gotten exactly the same thing when we integrated with
respect to x that we got when we integrated with respect to y

Therefore this equation,

 

must be identical with this equation



The equations will be identical if and only if

 and 

Therefore the general solution is



Edwin