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How can the following differential equation can be solved? $ \frac{dy}{dt}=3+e^{-t} -\frac{1}{2}y $ I proceeded by by rearranging the equation as follows $ \frac{dy}{dt}+\frac{1}{2}y=3+e^{-t} $My idea was to make the LHS a derivatives of two variables so that it could be integrated. But apparently I could not do that. How should i proceed now?
Your help is much appreciated.Thankyou.

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    @nihilisticgeek: please see: Earl A. Coddington: _An Introduction to Ordinary Differential Equations_, p39. It can be very useful.2011-07-25

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You need to use what's called an integrating factor. Since the coefficient of $y$ is simply the constant $1/2$, the factor is simple: $\mu = e^{\int 1/2 dt} = e^{t/2}$. If you multiply both sides of the differential equation by $\mu$, you can "factor" the left-hand side as an implicit differentiation like so:

\mu y' + 1/2\cdot\mu y = \mu\cdot(3+e^{-t}); (\mu y)' = \mu\cdot(3+e^{-t}); (e^{t/2} u)' = 3e^{t/2}+e^{-t/2}.

This can be seen with the product rule and because of the fact we chose $\mu$ so that \mu' = 1/2 \cdot\mu.

From here you can integrate both sides and then isolate the function $y$,

$ e^{t/2} y = 6e^{t/2} -2e^{-t/2}+C;$ $ y = 6-2e^{-t}+Ce^{-t/2}. $

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    If that's what you got as a final answer, then your error was that you differentiated $3e^{t/2}+e^{-t/2}$ instead of integrated.2011-07-23