Question

dy/dt=-y+5 y(0) =a

Answer 1

$$\begin{array}{l} \frac{dy}{dt} = -y + 5, \quad y(0) = a \\ \frac{dy}{dt} = -y + 5 \Rightarrow \frac{dy}{dt} = -(y - 5) \Rightarrow \frac{dy}{y - 5} = -dt; \\ \int \frac{dy}{y - 5} = -\int dt; \\ \ln |y - 5| = -t + C, \end{array}$$

where $C = \text{const}$.

So $\ln |y - 5| = -t + C \Rightarrow y - 5 = e^{C - t} \Rightarrow y = 5 + e^{C - t}$.

As $y(0) = a$ then $a = 5 + e^{C - 0} \Rightarrow C = \ln |a - 5|$.

Therefore

$$y = 5 + e^{\ln |a - 5| - t} = 5 + \frac{a - 5}{e^t}$$

Answer: $y = 5 + \frac{a - 5}{e^t}$