Consider the differential equation dy/dt=y-t

a) Determine whether the following functions are solutions to the given differential
equation.
y(t) = t + 1 + 2e^t
y(t) = t + 1
y(t) = t + 2
b) When you weigh bananas in a scale at the grocery store, the height h of the
bananas is described by the differential equation d^2h/dt^2=-kh
where k is the spring constant, a constant that depends on the properties of
the spring in the scale. After you put the bananas in the scale, you (cleverly)
observe that the height of the bananas is given by h(t) = 4 sin(3t). What is the
value of the spring constant?

take the derivative of each

a. y'=1+2e^t
b. y'=1
c. y'=1

now put in for y', y-t
looks like a works.

b. h=4sin3t
h'=12cos3t
h"=-36sin3t=-kh=-k*4sin3t
k=9

check that.

I actually got 36 for K. Is this right? =4sin(3t)

h'= 4*3*cos(3t) = 12cos(3t)
h"= 12*3*(-sin(3t)) = -36sin(3t)
h"= -k*h = -36*sin(3t)
=36

a) To determine if the given functions are solutions to the differential equation dy/dt = y - t, we need to substitute the functions into the equation and check if they satisfy it.

Let's start with y(t) = t + 1 + 2e^t:
dy/dt = 1 + 2e^t
Substituting into the differential equation:
1 + 2e^t = t + 1 + 2e^t - t
1 = 1
Since the equation holds true, y(t) = t + 1 + 2e^t is a solution to the differential equation.

Next, let's check y(t) = t + 1:
dy/dt = 1
Substituting into the differential equation:
1 = t + 1 - t
1 = 1
Again, the equation holds true, so y(t) = t + 1 is also a solution to the differential equation.

Finally, y(t) = t + 2:
dy/dt = 1
Substituting into the differential equation:
1 = t + 2 - t
1 = 2
This equation does not hold true, so y(t) = t + 2 is not a solution to the differential equation.

b) To find the value of the spring constant k in the differential equation d^2h/dt^2 = -kh, we can use the given height function h(t) = 4sin(3t).

First, differentiate h(t) twice with respect to t:
dh/dt = 4(3)cos(3t) = 12cos(3t)
d^2h/dt^2 = -36sin(3t)

Substituting these derivatives into the differential equation:
-36sin(3t) = -k(4sin(3t))

We can see that sin(3t) is common on both sides of the equation, so we can cancel it out:
-36 = -4k

Finally, solve for k:
k = -36 / -4
k = 9

Therefore, the value of the spring constant k is 9.