# Proper constants for $\alpha, \beta$

Here is the problem:

For what values of $\alpha$ and $\beta$, the function

$$\mu(x,y)=x^{\alpha}y^{\beta}$$

is an integrating factor for the OE $$ydx+x(1-3x^2y^2)dy=0.$$ I am working on it just knowing the definition. :(

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Let $$M(x,y)dx+N(x,y)dy=0$$

If exists $F$ such that $$dF=M(x,y)dx+N(x,y)dy$$

then

$$F(x,y)=C$$

for some $C \in \mathbb{R}$ defines an implicit solution of the ODE

When does $F$ exists? Exists if

$$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$

and $$M(x,y)=\frac{\partial F}{\partial x}, N(x,y)=\frac{\partial F}{\partial y}$$

Multiply your equation by $\mu(x,y)=x^{\alpha}y^{\beta}$

$$x^{\alpha}y^{\beta+1}dx+x^{\alpha+1}y^{\beta}(1-3x^2y^2)dy=0$$

Identify $M(x,y)$ and $N(x,y)$ and apply the above theory. I found that $\alpha=\beta=-3$

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If your OE is of form $$yf(xy)dx+xg(xy)dy=0$$ then you can assume that the integrating factor is $$\mu(x,y)=\frac{1}{xy(f-g)}$$ It means you have it here $$\mu=\frac{1}{xy(1-1+3x^2y^2)}$$