Question

Find the general solution to the differential equation modeling how a person learns:
Dy/dt=100-y. Then find the particular solutions with the following initial conditions:
y(0)=35: y=
y(0)= 125: y=
Plot the slope field of this differential equation and sketch the solutions with y(0) = 35 and y(0) = 125. Which of these two particular solutions could represent how a person learns?
A. y(0) =35
B. y(0) = 125.
C. either of these
D. none of the above

Answer 1

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 person can learn (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 D: none of the above.

If you want to learn more, you can read:

https://brainly.com/question/353770