When considering the general form (which is an initial value problem)
$$\frac{{dy}}{{dt}} = ay – b
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with initial condition y(0)=y0 (Where y0 is an arbitrary initial value)
If
$$\begin{array}{l}a \ne 0\\y \ne \frac{b}{a}\end{array}
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The testbook I have rewrites the general form as:
$$\frac{{\frac{{dy}}{{dt}}}}{{y – (\frac{b}{a})}} = a
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$$
I don't understand why they would rewrite in this way. The only connection I can make in my mind that the derivative is related to the limit which 1/0 would be undefined or a condition associated with a limit. Any insight that some one can provide for this rewrite would really clear up a lot for me.
This leads to
a solution of the initial value problem of
$$y = (\frac{b}{a}) + [y0 – (\frac{b}{a})]{e^{at}}
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$$
Thanks in advance.
Best Answer
The reason that the ODE is written as you showed is so that terms containing $y$ are on one side of the equation only and terms not containing $y$ are also on one side of the equation only. Rewriting that way allows you to integrate both sides of the equation, the left hand side with respect to $y$ and the right hand side with respect to $t$. The left hand side becomes $\ln (y - \frac{b}{a})$, while the right hand side becomes $at + C$, where $C$ is some constant.