Why is it that when you have the differential equation y\'[x]=r y[x], and you ch

Question

Why is it that when you have the differential equation y'[x]=r y[x], and you change the sign (from negative to positive or vice versa) of r, you end up changing the function from exponential growth to decay? Note: y[x]=k e^rx Example: y[x]=k e^rx = exponential growth for r>0 Example: y[x]=k e^-rx = exponential decay for r>0 Also, Why is it that when you have the differential equation y'[x]=r y[x], and you change the sign (from negative to positive or vice versa) of k, you end up changing the function from exponential growth to decay? Note: y[x]=k e^rx Example: y[x]=k e^rx = exponential growth for k>0 Example: y[x]=-k e^rx = exponential growth for k>0 Please explain these concepts in enough detail to get your point across effectively. Thanks!!

Explanation / Answer

Why is it that when you have the differential equation y'[x]=r y[x], and you change the sign (from negative to positive or vice versa) of r, you end up changing the function from exponential growth to decay?

Because "r" stands for your rate; in the solved form of the diff. eq., it always ends up as the power of the e - in other words, you always end up with er. When it's negative, that's going to be 1/er and the derivative will be negative. A function with a negative derivative will be decreasing; a function with a positive derivative (so just er) will be increasing.

Hence r > 0 is growth and r < 0 is decay.

Note: y[x]=k e^rx Example: y[x]=k e^rx = exponential growth for r>0 Example: y[x]=k e^-rx = exponential decay for r>0

Also, Why is it that when you have the differential equation y'[x]=r y[x], and you change the sign (from negative to positive or vice versa) of k, you end up changing the function from exponential growth to decay?

Same idea - k, in the solved form, will become ek, determing whether there is growth or decay by virtue of being e's "power."

Note: y[x]=k e^rx Example: y[x]=k e^rx = exponential growth for k>0 Example: y[x]=-k e^rx = exponential growth for k>0 Please explain these concepts in enough detail to get your point across effectively. Thanks!!

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