Krdytkyu

I'm trying to learn to solve differential equations. Currently in class we are discussing integration factors. Here is one of the problems on the homework -

$$(3xe^y+2y)dx + (x^2e^y+x)dy=0tag{1}$$

We can quickly see that if we take $frac{partial}{partial y}$ of the left term and $frac{partial}{partial x}$ of the right term and we see that they are not equal, so we need an integrating factor.

An integrating factor of $x$ gets us a bit farther:

$$(3x^2e^y+2xy)dx + (x^3e^y+x^2)dy=0tag{2}$$

And now when we take $frac{partial}{partial y}$ of the left term and $frac{partial}{partial x}$ of the right term , the two are equal.

From here, I think we would integrate the left term with respect to $x$ and right term with respect to $y$ as follows -

$$int {(3x^2e^y+2xy)dx} + int (x^3e^y+x^2)dy = 0 tag{3}$$
$$2x^3e^y+2x^2y+C=0tag{4}$$
$$x^3e^y+x^2y=Ctag{5}$$

Am I thinking about this the right way?

Also, in $(5)$, is it fine to move the Constant to the right and absorb the common coefficient of 2 into it to simplify the expression?
$(5)$ is the answer in the book and (4) was my answer. Either I have done something wrong, or I am confused in regard to what may be combined into the constant $C$.

$$(3xe^y+2y)dx + (x^2e^y+x)dy=0tag{1}$$
multiply by X as integrating factor as you wrote
$$(3x^2e^y+2yx)dx + (x^3e^y+x^2)dy=0$$
Rearrange terms
$$(3x^2e^ydx+x^3e^ydy)+(2yxdx + x^2dy)=0$$
$$d(x^3e^y)+d(x^2y)=0$$
After integration
$$x^3e^y+x^2y=K$$
For the constant yes it absorbs the factor 2 and you can have it on the other side. Your result is correct but I wouldn't write the integrals you wrote...
$$2x^3e^y+2x^2y+C=0$$
$$implies x^3e^y+x^2y=-frac C 2$$
Substitute
$$K=-frac C2 implies x^3e^y+x^2y=K$$
K is just a constant.