For example:
[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} ?
