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comiconor · Jun 10, 2014, 12:40 PM

If I have an equation in the form: dy/dx + f(x)y = g(x)

How can it be solved as y = e^-w * (integral (g(x)*e^w dx) + k) ?

To make it simpler, g(x) = 0, which would make it:

y = e^-w * k

But I still don't see how you arrive at this formula.

comiconor · Jun 10, 2014, 12:41 PM

Just to be clear, w = integral f(x) dx

ebaines · Jun 10, 2014, 12:58 PM

Seems to work. If g(x) = 0 then:

$\frac{dy}{dx}+f(x)y=0$

If $y(x)=Ke^{-w(x)}$ , then substituting in the above yields:

$-Ke^{-w(x)}\frac{dw}{dx}+f(x)Ke^{-w(x)}$

We can substitute for $\frac{dw}{dx}=\frac{d(int(f(x))dx}{dx}=f(x)$ to get:

$-Ke^{-w(x)}f(x)+f(x)Ke^{-w(x)}=0$ , as expected.