# Integrating factor in Exact Equations

#### alane1994

##### Active member
Ok, so I have this differential equation.

$(3x^2y+2xy+y^3)+(x^2+y^2)y\prime=0$

First I needed to check to see if it is exact.

$$M=3x^2y+2xy+y^3$$
$$N=x^2+y^2$$

$$\dfrac{\partial M}{dy}(3x^2y+2xy+y^3)=3x^2+2x+3y^2$$

$$\dfrac{\partial N}{dx}(x^2+y^2)=2x+0$$

For the integrating factor, I believe it is in the form,

$\dfrac{M_y(x,y)-N_x(x,y)}{N(x,y)}\mu$

And so, I would have,

$$\dfrac{3x^2+2x+3y^2-2x}{x^2+y^2}\mu = \dfrac{3x^2+3y^2}{x^2+y^2}\mu=\dfrac{d\mu}{dx}$$

Now... I feel as though I am wrong in this. Could someone look and tell me if I am on the right track, and if I am off guide me back to the path of righteousness?

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#### Opalg

##### MHB Oldtimer
Staff member
And so, I would have,

$$\dfrac{3x^2+2x+3y^2-2x}{x^2+y^2}\mu = \dfrac{3x^2+3y^2}{x^2+y^2}\mu=\dfrac{d\mu}{dx}$$

Now... I feel as though I am wrong in this. Could someone look and tell me if I am on the right track, and if I am off guide me back to the path of righteousness?
You seem to be on the right track, but you need to go a little bit further, by noticing that $\dfrac{3x^2+3y^2}{x^2+y^2} = \dfrac{3(x^2+y^2)}{x^2+y^2} = 3.$ #### alane1994

##### Active member
$$\dfrac{d\mu}{dx}=3\mu$$

Would $$\mu(x)=0$$?

#### Opalg

##### MHB Oldtimer
Staff member
$$\dfrac{d\mu}{dx}=3\mu$$

Would $$\mu(x)=0$$?
No. Think again (think exponential).

#### Chris L T521

##### Well-known member
Staff member
$$\dfrac{d\mu}{dx}=3\mu$$Would $$\mu(x)=0$$?
Sure, but things would be boring if that was the only solution. Instead, try solving the ODE by separation of variables.

EDIT: Ninja'd by Opalg.

#### alane1994

##### Active member
I feel foolish, I am unsure how to get what is required.
I typed it into wolfram and got,
$$\mu(x)=e^{3x}$$

#### alane1994

##### Active member
$$\dfrac{d\mu}{dx}=3\mu$$
$$\dfrac{1}{\mu}\dfrac{d\mu}{dx}=3$$
$$\dfrac{1}{\mu}d\mu=3dx$$
Then solve yeah?

#### Opalg

##### MHB Oldtimer
Staff member
$$\dfrac{d\mu}{dx}=3\mu$$
$$\dfrac{1}{\mu}\dfrac{d\mu}{dx}=3$$
$$\dfrac{1}{\mu}d\mu=3dx$$
Then solve yeah?
Yes! #### alane1994

##### Active member
$$C=(3x^2y+y^3)e^{3x}$$

EDIT: Is this correct?

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