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Integrating Factor

MathJax TeX Test Page I think a good way to preface this section is to recall the product rule from single variable calculus. $$\boxed{\dfrac{d}{dx}\left(f(x)g(x)\right) = f'(x)g(x) + f(x)g'(x)}$$ With that in mind, let's look at an example: $$\dfrac{dy}{dx} - 3y = e^{2x}, y(0) = 3$$ This isn't exactly the product rule, but it can be. Let's multiply by a function, $\rho$. $$\rho\dfrac{dy}{dx} -3\rho{}y = \rho{}e^{2x}$$ So, on the left side, the y is differentiated, so on the right, the $\rho$ is differentiated. We can say $$\dfrac{d\rho}{dx} = -3\rho \to \rho = e^{-3x} + C$$ We can pick our own function, so we just set C = 0. So, what we were trying to do is combine the left side of the equation into the derivative of one term. This is what it becomes: $$\dfrac{d}{dx}(e^{-3x}y) = e^{-x}$$ Now we just integrate, and we're done. Here, we can't forget the constant term. Before, we were allowed to, because we just had to find any function that satisfy the product rule. $$e^{-3x}y = -e^{-x} + C \to \boxed{y = Ce^{3x} - e^{-x}}$$
David Witten

Homogeneous Equations with Constant Coefficients

Exact Equations