Calculus

You use the integrating factor when the differential equation is not exact. 1. First Order Linear Ordinary Differential Equations If the differential equation is in the form $\frac{\mathrm{d} y}{\mathrm{d} x}+f(x)\cdot y=g(x)$, you could use the integrating factor, $\mu(x)$. Keep this form in mind. Remember the product rule? It’s $\frac{\mathrm{d} }{\mathrm{d} x}(f\cdot g)=f’\cdot g+f\cdot g’$. Let’s replace $f$ with the $y$ and $g$ with a function that we don’t know $\mu(x)$, so that it satisfies $\frac{\mathrm{d} }{\mathrm{d} x}(\mu(x)\cdot y)=\mu’(x)\cdot y+\frac{\mathrm{d} y}{\mathrm{d} x}\cdot \mu(x)$, so that you can get a form that’s similar with $\frac{\mathrm{d} y}{\mathrm{d} x}+f(x)\cdot y=g(x)$....