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The DE is:

dy/dt = e^(-t/20) - (1/40)y

I moved the last term to the left, giving dy/dt + (1/40)y = e^(-t/20). I had e^(t/40) as my integrating factor. Going in my (wrong) direction, I was going to

end up with y = 1+ce^(-t/40). The book gets Y(t) = -40e^(-t/20) + Ce^(-t/40).

Where I am confused is the -40e^(/t-20). I thought when multiplying thru by the IF that I would get a "1" on that side.

What am I missing?

Thank you!