find the solution of the differential equation dy/dt = ky, k a constant, that satisfies the given conditions:
y(0)=50, y(5)=100
please show steps.
INtegrate dy/dt
I think the answer will be in the form of y=e^kt + C
check that and see if it works.
You should be memorizing your integral forms.
but how do i find k?
To find the constant k, we can use the initial condition y(0) = 50.
We know that y(t) = e^(kt) + C. Plugging in the initial condition y(0) = 50 into this equation gives us:
50 = e^(k*0) + C
Simplifying, we have:
50 = 1 + C
Therefore, C = 50 - 1 = 49.
So, the general solution to the differential equation is:
y(t) = e^(kt) + 49
Now we need to find the value of k using the second initial condition y(5) = 100.
Plugging this condition into the general solution, we have:
100 = e^(k*5) + 49
Subtracting 49 from both sides, we get:
51 = e^(k*5)
To solve for k, we need to take the natural logarithm of both sides:
ln(51) = ln(e^(k*5))
Using the property of logarithms, we can bring the exponent down:
ln(51) = k*5
Now, divide both sides by 5 to solve for k:
k = ln(51) / 5
Thus, the value of k is ln(51) / 5.