# What is the integrating factor for #0 = (3x^2 + 3y^2)dx + x(x^2 + 3y + 6y)dy#?

I got:

Two options to find the special integrating factor, as defined by Nagle, are:

for the differential

For now, let's find the partial derivatives. For your differential:

Therefore:

and:

Checking for exactness, we obtain:

so we know our integrating factor is correct!

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The integrating factor for the given differential equation (0 = (3x^2 + 3y^2)dx + x(x^2 + 3y + 6y)dy) is (e^{\int P(x) , dx}), where (P(x)) is the coefficient of (dy) after dividing the equation by (dx). In this case, (P(x) = \frac{x(x^2 + 3y + 6y)}{3x^2 + 3y^2}).

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The integrating factor for the given first-order linear ordinary differential equation (0 = (3x^2 + 3y^2)dx + x(x^2 + 3y + 6y)dy) is (\mu(x, y) = e^{\int P(x)dx}), where (P(x)) is the coefficient of (dy) after rearranging the equation into the standard form (M(x, y)dx + N(x, y)dy = 0). In this case, (P(x) = \frac{dN}{dy} - \frac{dM}{dx}). After calculation, we find (P(x) = 1), so the integrating factor is (\mu(x, y) = e^{\int dx} = e^x).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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