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[tex]\frac{dy}{dx} + y = e^{3x}[/tex]

I understand that these differential equations are most easily solved by multiplying in a factor of integration, and then comparing the equation to the product rule to solve et al..

For example:

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

[tex]\frac{dy}{dx} + 2ty = t[/tex]

Multiplying in an integration factor u(x), which in this case:

[tex]u(x) = e^{\int{2t}dt} = e^{t^{2}}[/tex]

[tex]e^{t^{2}}\frac{dy}{dx} + 2te^{t^{2}}y = te^{t^{2}}[/tex]

Now I can compress the left side down using the product rule and all that.

I don't understand how they are getting u(x) or why it's equal to e^{\int{2t}dt} ?

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# First order non-separable linear deq's using an integration factor?

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