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anonymous
 one year ago
How do you reconcile where d & c are constants?
y[x] = d E^( r x  c r)
Reconcile with the formula
y[x] = k E^(r x)
anonymous
 one year ago
How do you reconcile where d & c are constants? y[x] = d E^( r x  c r) Reconcile with the formula y[x] = k E^(r x)

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I cant seem to work out how you would get c out of the exponent?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0its easy if c is defined to something, but if it's just c.. ??

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.1c and d are constants... well using product rule that d is going to go bye bye if you leave the second term alone and deal with the first

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0or do you leave c in the exponent?

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.1is r the variable in this case?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0but does taking the derivative mean to 'reconcile'?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ah, the actual question went.. When you start with y'[x] = r y[x] , give it one data point, and solve for y[x] , you always get something of the form y[x] = k e^(r x) where k is a constant determined by the data point. Try it with the data point y[c] = d where c and d are constants:

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0This gives you y[x] = d e^((r) c + r x) . How do you reconcile this output with the formula y[x] = k e^(r x) above?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0It looks like d and c need to combine somehow to form k ?

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.1no.. and ugh sorry for the wait my batteries died on my wireless mouse

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0y'[x]/y[x] = r is there a formula for just k?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I get y[0] = d E^(cr)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so that would be k, right?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0y[x] = d e^(cr + r x) . y[x] = d e^(cr) E^(r x) . y[x] = d/e^(cr) E^(r x) . k = d/e^(cr) r = r

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0let me know if Im off mark.. thanx.. hitting the sack

phi
 one year ago
Best ResponseYou've already chosen the best response.5I would analyze it this way \[y'[x] = r y[x] \\ \frac{dy}{dx} = r y \\ \frac{dy}{y}= r dx \\ \int \frac{dy}{y}= \int r\ dx\] now integrate, and explicitly note the constant of integration: \[ \ln y +c_1= rx + c_2 \\ \ln y = r x +(c_2c_1) \\ \ln y = r x + c\] I put in a constant on both sides, but as you can see, because they are arbitrary, you can combine them into a single arbitrary constant. Most people would just write the constant on one side, like this \[ \int \frac{dy}{y}= \int r\ dx \\ \ln y = r x + c \] now make each side the exponent of "e" \[ y = e^{rx+c}\\ y= e^c\ e^{rx} \] "c" is an arbitrary number, so that means e^c is also some number. Call it k. Thus \[ \ y= k\ e^{rx} \]

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.1that was awesome @phi I think it was due to @hughfuve 's notation on the equation that got me really confused so I couldn't go on further, but when you've changed it, I can understand that we need to use separation of variables

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0wow that's awesome hey. separation of variables and integration

phi
 one year ago
Best ResponseYou've already chosen the best response.5to finish up the details data point y[c] = d where c and d are constants: this means when x=c, y = d, i.e. point (c,d) is a solution to the equation \[ y = k\ e^{rx} \] plug in x=c , y=d and solve for k: \[ d= k\ e^{rc} \\ k = d\ e^{rc} \] notice this matches with *** How do you reconcile where d & c are constants? y[x] = d E^( r x  c r) Reconcile with the formula y[x] = k E^(r x) *** k= d E^(cr) (some constant determined by the fixed constants c , d and r)
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