Final answer:
The differential equation y'(t) = e^(5t - y) with the initial condition y(0) = 0 is solved by separating variables, integrating, and then applying the initial condition to find the integration constant. The final explicit solution for y(t) is y = ln(1/5 * e^(5t) + 4/5).
Step-by-step explanation:
The task is to solve the differential equation y'(t) = e^(5t - y) with the initial condition y(0) = 0. This is a separable differential equation since we can isolate all the y terms on one side and all the t terms on the other side.
First, we rearrange the equation to separate variables:
\(\frac{dy}{dt} = e^{5t}e^{-y}\). Then, we take \(e^y\) to the other side to get
\(e^y dy = e^{5t} dt\). Now, we can integrate both sides:
\(\int e^y dy = \int e^{5t} dt\), leading to
\(e^y = \frac{1}{5}e^{5t} + C\).
Applying the initial condition y(0) = 0, we find that
\(e^0 = \frac{1}{5}e^0 + C\) which simplifies to
1 = \frac{1}{5} + C, giving us
C = \frac{4}{5}. Hence,
\(e^y = \frac{1}{5}e^{5t} + \frac{4}{5}\).
To solve for y explicitly, we take the natural logarithm of both sides:
\(y = ln\left(\frac{1}{5}e^{5t} + \frac{4}{5}\right)\). This is the solution in explicit form.