Hello wonderful mathematics people, this is Anna Cox from Kellogg Community College exponential change and separable differential functions The law of exponential change states that y equal y naught, e to the kt where k greater than zero means growth and k less than zero is decay and k is the rate constant of the equation y naught is the initial amount and y is the final amount after time t Differential equation, if we have dy dt equal ky, we're gonna put all of the y's on one side and everything else on the other so we're gonna end up with dy divided by y equaling k dt And now we're gonna integrate both sides. So the integral of one over y dy is gonna be the natural log of the absolute value of y The integral of k dt, remember k is a constant, so that's just gonna be kt plus some constant c. Now that c could come on either side, we want to have it on the right so that when we take e, e to each side to the base e so if I have natural log, the absolute value of y equaling kt plus c. If I take each side to the e power that e to the ln is gonna cancel leaving us the absolute value of y That's gonna equal e to the kt plus c. Now a different way to think of that would be just to change the logarithmic form into exponential form So whenever we have two things in the exponent that are added we can think about separating those as each of the bases being multiplied together so we get y equal absolute value of y equaling, e to the kt e to the c, where e to the c is some constant We call that constant A, so we get y equaling some constant A e to the kt Now we can drop the absolute value because e to any number, e to any c is gonna be positive and e to the kt is also gonna be positive Next we're gonna think about what happens when t equals zero, so if t equals zero we're gonna realize that we have the final value and the original values gonna be the same time zero so y naught equal A e to to the k zero. Anything to the zero power is one so our y naught is really just A So y equal y naught e to the kt. Sometimes you'll see it as y equal Ae to the kt, where A is that initial value Number of cases of disease is reduced by 20% per year If 10,000 cases today how long for a thousand cases? So we're gonna use this y to the y naught e to the kt equation if we're reduced by 20% it means that we have 80% left. So 0.80 of our ten thousand original equals our 10,000 e to the k, and this is per year, so one is our time We're gonna find our constant, so we're gonna start by dividing each side by 10,000 Reducing it we get four fifths, change the exponential into logarithmic, so we get k equaling the natural log of four fifths Now once we find k it's gonna be true for every single problem that uses this information So now we wanna know how long, so we're trying to find the time for a thousand cases. So if we have a thousand equaling the original amount of 10,000 e to the k. We found the k a moment ago: ln four fifths times t And do the same steps we're gonna divide by 10,000 We're gonna get one tenth. We have e ln four fifths times t, but that t could go up into the exponent by the power rule for logarithms so that this e to the ln would cancel so we get one tenth equaling four fifths t If one tenths equal four fifths t Take the natural log of each side. Natural log of one tenth equal natural log of four fifths Bring the t down in front and divide each side by ln four fifths So ln one tenth divided by ln four fifths is about 10.32 years We can do problems that are involving interest also so the interest paid by compound in continuously A equal A naught e to the rt or sometimes it's thought of P equal P naught e to the rt Pert Shampoo is how we frequently have students try to help themselves remember this. PERT the Pert shampoo Radioactive decay, the k is gonna be a negative when we have decay, so we could think of it as a negative of a positive so if k was greater than zero Some books talk about y equal y naught e to the kt where k is less than zero. Just want you to be familiar that it could be written either way and it's dependent on whether we're putting a stipulation k is greater than or k is less than zero If we're doing a half life problem Half life would mean that we end with half of what we started with So one half of y naught would equal y naught e to the kt The y naughts would divide out and we'd get one half equal e to the kt putting taking the natural log of each side so the natural log of half, natural log e to the kt The natural log of e are gonna cancel, so we get kt on the right side One half we could think of as ln to the two to the negative one power and then we can bring that negative one down in front So negative ln two is gonna equal kt. Solving for k, we can get the constant is negative ln two over t or solving for time, time equal negative ln two over k. So we either need a constant or the time to be able to figure out um, the relationship on a half life Newton's law of cooling, H is temperature of an object, t is time, Hs is the temperature in the surrounding area So, dH dt equal negative k H minus Hs So we're gonna start by letting y equal H minus Hs If y equals that then we take the derivative in terms of time, so we get dy dt equal d dt of H minus Hs or dy dt equal dH dt minus dHs dt Now this Hs is the temperature of the surroundings So it's a constant, the temperature of the surroundings not changing and we know that the derivative of any constant is just zero So dy dt equals dH dt So dy dt, from up here is just negative k H minus Hs, because we decided that errr , we showed the dH dt was really just dy dt. So by our transitive properties those three things have to all equal each other Now we're gonna take and put all the y's on one side That H minus Hs we said was y, so we're gonna have dy over y equaling negative k dt. Taking the integral of both sides to show look like what we did in the beginning So we'd end up with the natural log of the absolute value of y equaling negative kt plus some constant changing that into exponential form we get y equal e to the negative kt plus c separating that up so we have e to the c, e to the c we're just gonna call some constant A So when t equals zero we know that the original and the ending is the same amount So y would equal y naught when time a zero. Solving for that, we'd get A equal y naught. This is the equation we had before So y equal y naught e to the negative kt, we've basically just proven it twice Now since H minus Hs equal y, we can see that H minus Hs equal H naught minus Hs e to the negative kt That's Newton's Law of Cooling Thank you and have a wonderful day!