I'm confused as to how to find the arbitrary constant using the initial condition given.
This is the equation I have to solve,
$\frac{dy}{dx}=a-by$, with initial condition $y(0)=0$. Also, a and b are constants.
I've used integration by separation to get,
$a-by=Ae^{-bx}$, where $A=e^c$.
I'm not sure what to do next?
Best Answer
$$y'(x)=\text{a}-\text{b}y(x)\Longleftrightarrow\int\frac{y'(x)}{\text{a}-\text{b}y(x)}\space\text{d}x=\int1\space\text{d}x\Longleftrightarrow-\frac{\ln\left|\text{a}-\text{b}y(x)\right|}{\text{b}}=x+\text{C}$$
So, when $y(0)=0$:
$$-\frac{\ln\left|\text{a}-\text{b}\cdot0\right|}{\text{b}}=0+\text{C}\Longleftrightarrow\text{C}=-\frac{\ln\left|\text{a}\right|}{\text{b}}$$
So, we get:
$$-\frac{\ln\left|\text{a}-\text{b}y(x)\right|}{\text{b}}=x-\frac{\ln\left|\text{a}\right|}{\text{b}}\Longleftrightarrow\left|\text{a}-\text{b}y(x)\right|^{-\frac{1}{\text{b}}}=e^x\left|\text{a}\right|^{-\frac{1}{\text{b}}}$$