# What is the general solution of the differential equation? # (3x^2+2y)dx+2xdy=0 #

This is a First Order DE of the form:

Which we know how to solve using an Integrating Factor given by:

And so our Integrating Factor is:

If we multiply by this Integrating Factor we will (by its very design) have the perfect differential of a product:

Which is now a separable DE, and we can separate the variables to get:

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An alternative approach

to verify this we check the mixed partials

We then look at the other partial that follows by now differentiating this f wrt y:

Final bit of algebra:

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The given differential equation is ( (3x^2 + 2y)dx + 2xdy = 0 ).

To find its general solution, we rearrange it into the form ( M(x,y)dx + N(x,y)dy = 0 ).

Comparing the equation with ( M(x,y)dx + N(x,y)dy = 0 ), we have ( M(x,y) = 3x^2 + 2y ) and ( N(x,y) = 2x ).

To check for exactness, we compute the partial derivatives:

[ \frac{\partial M}{\partial y} = 2 ] [ \frac{\partial N}{\partial x} = 2 ]

Since ( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} ), the equation is exact.

To find the general solution, we integrate ( M ) with respect to ( x ) and ( N ) with respect to ( y ):

[ \int M(x,y) , dx = \int (3x^2 + 2y) , dx = x^3 + 2xy + f(y) ] [ \int N(x,y) , dy = \int 2x , dy = 2xy + g(x) ]

Since ( x^3 + 2xy + f(y) = 2xy + g(x) ), we equate the like terms to find ( f(y) ) and ( g(x) ):

[ f'(y) = 2 \Rightarrow f(y) = 2y + C_1 ] [ g'(x) = x^3 \Rightarrow g(x) = \frac{1}{4}x^4 + C_2 ]

Thus, the general solution is given by:

[ x^3 + 2xy + 2y = \frac{1}{4}x^4 + C ]

where ( C = C_1 + C_2 ) is the constant of integration.

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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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