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## Main Question or Discussion Point

To solve a separable ODE like this I would simply multiply each side by dx and then integrate both sides. However, I know that it is only notational convenience that allows me to do this, and what's really going on is slightly more complicated.

Take this DE for example:

[tex]y^{2}\frac{dy}{dx}=x+1[/tex]

[tex]\int{(y^{2}\frac{dy}{dx})}dx=\int{(x+1)}dx[/tex]

[tex]\int{y^{2}}dy=\int{(x+1)}dx[/tex]

[tex]\frac{y^{3}}{3} = \frac{x^{2}}{2} + x + c[/tex]

What I don't understand, is how the LHS simplified like this:

[tex]\int{(y^{2}\frac{dy}{dx})}dx = \int{y^{2}}dy[/tex]

I'm sorry for asking such a basic questions, but my book does not explain this well at all :(

Take this DE for example:

[tex]y^{2}\frac{dy}{dx}=x+1[/tex]

[tex]\int{(y^{2}\frac{dy}{dx})}dx=\int{(x+1)}dx[/tex]

[tex]\int{y^{2}}dy=\int{(x+1)}dx[/tex]

[tex]\frac{y^{3}}{3} = \frac{x^{2}}{2} + x + c[/tex]

What I don't understand, is how the LHS simplified like this:

[tex]\int{(y^{2}\frac{dy}{dx})}dx = \int{y^{2}}dy[/tex]

I'm sorry for asking such a basic questions, but my book does not explain this well at all :(