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enter image description here I know how to solve the exercise using the hint. But I do not understand where the hint is coming from. Is it just continous compounding?

Can anybody explain $f(t,r) = e^{at}r$? What does it stand for and where does it come from?

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$e^{at}$ is simply the Integrating factor since it reduces the problem to a differential for $f(t,r)$ which is easy to solve. The $a$ comes from the coefficient in front of $r(t)$ in your equation.

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