Question

The unique solution to the initial value problem

y' + y = g(t), y(0)=y0
is y(t) = 6e^(-4t) -4t -1 Determine the constant y0 and the function g(t)
y0=
g(t)

Answer 1

The constant y0 is 5 and the function g(t) is (6e^(-4t) - 6)/4.

Step-by-step explanation:

In the given initial value problem, y' + y = g(t), y(0) = y0, the unique solution is given as y(t) = 6e^(-4t) - 4t - 1.

To determine the constant y0, we substitute the initial condition y(0)=y0 into the solution:

y(0) = 6e^(-4*0) - 4*0 - 1 = 6 - 1 = 5. Therefore, the constant y0 is 5.

To determine the function g(t) that satisfies y0 = g(t), we isolate g(t) by rearranging the equation:

y0 = 6e^(-4t) - 4t - 1

5 = 6e^(-4t) - 4t - 1

4t = 6e^(-4t) - 6

g(t) = (6e^(-4t) - 6)/4