how do I separate this?

Question

xy'=4y
dx/dy=(x^2y^2)/(1+x)
dy/dx=e^(3x+2y)
(4y+yx^2)dy-(2x+xy^2)dx=0
Please I need an explanation about how to do this step by step I have a differential equations test on monday plese help me. thanks

Answer 1

The easiest way tends to be to get all of your dx and dy differentials into numerators, and then divide by everything that's out of place.

xy´ = 4y
x (dy/dx) = 4y
x dy = dx 4y

So at this point the x is with a dy, which we don't want, and the 4y is with a dx which we also don't want. So we can divide both sides by 4xy, or just xy:

(x dy)/(xy) = (4y dx)/(xy)
dy/y = 4/x dx

That's all you need to separate most times.

dx/dy = (x^2*y^2)/(1+ x)
dx = (x^2*y^2)/(1+ x) dy

We have a factor of x^2/(1+x) that is out of place, so we can divide both sides by this, which is usually expressed as dividing by x^2 and multiplying by (1+x), but if we look at it as just one big divide it keeps the "divide rule" consistant.

dx / (x^2/(1+x)) = [(x^2*y^2)/(1+x)] / (x^2/(1+x)) dy
(1+x)/x^2 dx = y^2 dy

Hopefully you get the picture - it's really straight-forward and basically just a form of algebraic manipulation.

dy/dx = e^(3x + 2y)
dy = e^(3x + 2y) dx
dy = e^(3x)*e^(2y) dx
--divide out the e^(2y)
1/e^(2y) dy = e^(3x) dx

(4y + y*x^2) dy - (2x + xy^2) dx = 0
y(4 + x^2) dx - x(2 + y^2) dx = 0
--divide both side by (4 + x^2)*(2 + y^2)
y/(2 + y^2) dy - x/(4 + x^2) dx = 0
y/(2 + y^2) dy = x/(4 + x^2) dx

Some of these examples introduced a need to be creative. With the e^(3x + 2y) example, I used the fact that x^(a+b) = x^a*x^b to break things up so I could divide. And in the last example, I just knew I wanted to be able to divide away, but you cannot divide by a single term (such as a binomial factor) that has both x and y in a single term, or you are putting a variable with the wrong differential. Dividing by two terms at one where each term has only x or y, such as dividing by (4 + x^2)*(2 + y^2) is fine, as this could be broken up into two separate divides, one of just (4 + x^2) and the other of (2 + y^2). I only do them at one for quicker manipulation. Dividing by xy in the first example is fine two because this could also be done in two separate divisions of first just x and then just y. You just want to make sure not to put an x in the denominator of you dy differential, or vice versa.

Hope that helps!

--charlie

Answer 2

#1: xy'=4y

This is easy, just divide by xy:

y'/y = 4/x

#2: dx/dy = x²y²/(1+x)

Divide by x²/(1+x) and multiply by dy:

(1+x)/x² dx = y² dy

#3: dy/dx = e^(3x+2y)

First, rewrite this as a product of functions of x and y:

dy/dx = e^(3x)e^(2y)

Multiply by dx and divide by e^(2y):

e^(-2y) dy = e^(3x) dx

#4: (4y+yx²) dy - (2x+xy²) dx = 0

This one is actually tricky. First, factor these functions into products of functions of x and y:

(4+x²) y dy - (2+y²) x dx = 0

Add (2+y²) x dx to both sides:

(4+x²) y dy = (2+y²) x dx

Now, the (4+x²) and (2+y²) are both on the wrong side, so divide by them to obtain:

y/(2+y²) dy = x/(4+x²) dx

In general, when dealing with separable differential equations, you will want to write the functions on both sides as the product of a function of x and a function of y. Then it should be easy to separate: divide by everything that contains an x and is on the dy side, and then divide by everything that contains a y and is on the dx side. This leaves you with only xs on the dx side and only ys on the dy side, so now you can integrate.