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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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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