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In an experiment, a certain type of bacteria was being added to a culture at the rate of e.03t+2e^{.03 t}+2 thousand bacteria per hour. Suppose that the bacteria grow at a rate proportional to the size of the culture at time t, with constant of proportionality k=.45. Let P(t) denote the number of bacteria in the culture at time t. Find a differential equation satisfied by P(t).

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Let y=P(y)y=P(y) be the number of bacteria in the culture at time tt. Since the constant of proportionality is k=.45k=.45, and that bacteria was added to a culture at the rate of e.03t+2e^{.03t}+2, then the differential equation that is satisfied by P(t)P(t) is

y=.45y+e.03t+2.\begin{aligned} \color{#4257b2}{y'=.45y+e^{.03t}+2}. \end{aligned}

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