How to find the integrating factor: dx + (x/y - sin y) dy = 0?

Question

Steps please. I understand the integrating factor is y but I don't know how it is determined. Thanks.

Answer 1

First, this equation is easier to solve for x, not y. So we have:

x' + x/y - sin y = 0

Putting this in standard form:

x' + 1/y x = sin y

Now, the integrating factor is, as you said, y. The reason we need y is because we want to take an equation of the form:

x' + f(y) x = g(y)

And turn it into a function of the form:

u(y) x' + u'(y) x = v(y)

Which can be integrated using the inverse product rule. So if we are to accomplish this by multiplying by an integrating factor u, we must choose u so that:

u'(y) = u(y) f(y)

So we must have:

u'(y)/u(y) = f(y)
ln (u(y)) = ∫f(y) dy
u(y) = e^(∫f(y) dy)

It is probably best to memorize this formula for the integrating factor, rather than rederiving it every time you use it. Also note that any antiderivative will enable the integration, we don't need to bother with finding the most general one. Anyway, returning to the actual equation:

x' + 1/y x = sin y

We use our formula for the integrating factor to find:

u(y) = e^(∫1/y dy) = e^(ln y) = y

Which was the integrating factor you need. Now multiplying:

y x' + x = y sin y

And indeed, this is a function of the form u(y) x'(y) + u'(y) x(y) = v(y). So now integrating both sides:

yx = ∫y sin y dy = - y cos y + ∫cos y dy = -y cos y + sin y + C

Dividing by y:

x = sin y / y - cos y + C/y

So we are done.

Answer 2

The equation at present is not exact, so we seek an integrating factor of the form (x^m)(y^n). The equation becomes (x^m)(y^n)dx + (x^m)(y^n)(x/y - sin y)dy = 0. Now we want the partial of (x^m)(y^n) wrt y to be the same as the partial of (x^m)(y^n)(x/y - sin y) wrt x. We find n(x^m)(y^(n-1)) = (m + 1)(x^m)(y^(n-1)) - m(x^9(m-1))(y^n)(sin y). Clearly, we should take m = 0 and m + 1 = n.

This gives us the integrating factor y, as you said. Now the equation is exact, and can be handled in the usual manner for exact equations.

Answer 3

dx + (x/y -sin y) dy = 0
=> y dx + x dy - ysin y dy = 0
=> d(xy) - y sin y dy = 0

Integrating,

xy - [ y integral sin y dy - intergal [ d/dy(y) integral siny dy ] dy = C
=> xy + y cos y - integral cos y dy = C
=> xy + y cos y - sin y = C

You asked this question again. Here
xdy + ydx = d(xy) is called the integrating factor.
It is to be found by observation.
For that you need to practise solving more of such problems.