Calc AB Need help. Separable Equations and solving.

Question

Consider the following.

y(5 + x) dx + x dy = 0

Determine whether the differential equation is separable.

I put separable

If the equation is separable, rewrite it in the form N(y) dy = M(x) dx.

 (If it is not separable, enter DNE in the answer blanks. Do not solve the differential equation.)

(1/y)dy=( ??? )dx

Answer 1

SEPARATES LIKE THIS:

y ( x+5)dx + x dy = 0

y ( x+ 5)dx = -x dy

(x+5) dx/(-x) = dy/y

[-1 - 5/x ] dx = dy/y

integrates:

-x - 5 ln x + c= ln y

exp (-x - 5 ln x + c)= y

k exp(-x) exp ( -5 ln x) = y

k exp(-x) exp ( ln x^(-5)) = y

-5k exp(-x) x^(-5) = y

C exp(-x) x^(-5) = y

Answer 2

y(5+x)dx+xdy=0

Divide both sides by xy:

[(5+x)/x]dx+(1/y)dy=0

Rearrange to get the desired form:

(1/y)dy=[-(5+x)/x]dx