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myrzilka [38]
2 years ago
12

Find the general solution to the differential equation modeling how a person learns:

Mathematics
1 answer:
Luda [366]2 years ago
3 0

Here we want to solve differential equations, we will see that the general solution is:

y = A*e^{-t} + 100

We want to solve the differential equation:

\frac{dy}{dt}  = 100 - y

From this is pretty clear that y is an exponential function, with an exponent of -1*t.

We can write it generally as:

y = A*e^{-t} + B\\\\\frac{dy}{dt}  = -A*e^{-t}

Then if we set B = 100 we get:

y = A*e^{-t} + 100\\\\\frac{dy}{dt}  = -A*e^{-t} \\\\\frac{dy}{dt}  = -A*e^{-t} - 100 + 100 = -y + 100

So we just found the general form of the function.

Now we have two cases:

A) y(0) = 35

y(0) = A*e^{-0} + 100  = 35\\=  A + 100 = 35\\\\A = 35 - 100 = -65

In this case, the function is:

y = -65*e^{-t} + 100

B) y(0) = 125

125 = A*e^0 + 100\\\\\125 - 100 = A\\\\25 = A

In this case, the function is:

y = 25*e^{-t} + 100

Now we want to see which one of the two can represent how a person learns. Just look at the graph below:

The green line is the one for y(0) = 35, and the blue one is for y(0) = 125.

Notice that for small values of t, the blue function is really large, thus it can't really model how a person learns (is larger for smaller values of t than for larger values).

So y(0) = 35 represents better how a <em>person can learn</em> (but not exactly, because you can see that it eventually becomes almost constant, which is something that really does not happen) so the correct option is <u>D: none of the above.</u>

If you want to learn more, you can read:

brainly.com/question/353770

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