Particular Solution

[Inspector Gadget]

Here is the problem...

Find the exponential function y=Ce^{kt} that passes through the points (0,4) and (5,1/2).

Just by looking at the problem, I know...

y=4e^{\frac{-3x \ln{2}}{5}}

...works, but how do you actually work out the problem? I've done it before given a point and the slope at that particular point, but I don't really know how to do this.

[philosophking]

y=Ce^(kt)

4=Ce^(k*0)
C=4

.5=4e^(k*5)
(1/8)=e^(5k)
ln(1/8)=5k
ln(1/8)/5 = k

Now that you have k and C, the problem's done!