I have the following 1st order linear differential equation:
$$\frac{dy}{dt}=ay+b$$
where $a$ and $b$ are constants.
A solution is using the following method to solve for $y$: $$\frac{dy}{dt}=a\left(y+\frac{b}{a}\right)$$ $$\frac{d}{dt}\left(y+\frac{b}{a}\right)=a\left(y+\frac{b}{a}\right)$$ $$y+\frac{b}{a}=Ce^{at}$$ $$y=-\frac{b}{a}+Ce^{at}$$
I have a hard time understanding how the 2nd equation can become the 3rd, and how the 3rd equation can become the 4th. What are the techniques that are used for the above method?
Thank you very much in advance!
$$\frac{dy}{dt}=a\left(y+\frac{b}{a}\right)$$ $$\frac{d(y+\frac ba)}{dt}=a\left(y+\frac{b}{a}\right)$$ Separate variables $$\frac{d(y+\frac ba)}{y+\frac ba}=adt$$ Integrate $$\int \frac{d(y+\frac ba)}{y+\frac ba}=a\int dt$$ $$\ln(y+\frac ba)=at+K$$ $$y+ \frac ba =Ce^{at}$$ $$y=- \frac ba +Ce^{at}$$