Linear First-Order Equations
The standard form is $\frac{dy}{dx}+P(x)y=Q(x)$. Multiply by the integrating factor to convert the left side into a derivative.
$$\text{I.F.}=e^{\int P(x)\,dx}$$
The standard form is $\frac{dy}{dx}+P(x)y=Q(x)$. Multiply by the integrating factor to convert the left side into a derivative.
Solve $\frac{dy}{dx}+3y=12$ if $y(0)=2$.
The integrating factor is $e^{3x}$.
Using $y(0)=2$, $C=-2$.
Final answer: $y=4-2e^{-3x}$.