## Related questions with answers

Question

As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, a.

Solve the differential equation.

Solution

VerifiedAnswered 2 years ago

Answered 2 years ago

Step 1

1 of 2Lets solve the equation

$\begin{align*} \dfrac{dy}{dt}&=-k(y-a) \\ \\ \dfrac{dy}{y-a}&=-kdt\tag{integrate both sides}\\ \ln (y-a)&=-kt+C\\ y-a&=Ce^{-kt}\\ y(t)&=a+Ce^{-kt}\\ \end{align*}$

For $t=0$, $y=1$.

$\begin{align*} 1-a&=C\\ y&=a+(1-a)e^{-kt} \end{align*}$

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