Find critical value of differential equation: 3y' - 2y = e^-"pi"t/2 , y(0) = a

[anami369]

Find critical value of differential equation: 3y' - 2y = e^-"pi"t/2 , y(0) = a

Hi guys! I am confused on how exactly do I do this problem:

Let a₀ be the value of a for which the transition form one type of behavior to another occurs. Estimate the value of a₀

3y' - 2y = e^-"pi"t/2 , y(0) = a

I know that y = (2 + a(3"pi"+4)e^2t/3 - 2e^-"pi"t/2)/(3"pi"+4)

How do I get started finding on finding a₀? T

Thanks in advance!

 

[HallsofIvy]

You say "transition form one type of behavior to another". What, exactly does that mean? What two different "types of behavior" could there be here? In particular, what is the difference in "behavior" of \(\displaystyle e^{2t/3"}\) and \(\displaystyle e^{-\pi t/2}\)?