Question

Last Homework problem! I cannot get this one! Please help!
The unique solution to the initial value problem,

Determine the constant y_0 and the function g(t)

Answer 1

The unique solution to the initial value problem
\[y'+y=g(t), y(0)=y_0\]
is y(t)=6e^(-2t)-4t+1
Determine the constant y_0 and the function g(t).

Answer 2

This is a day late, but hopefully this is still of use to you:

To solve for y_0, we can y(0) directly into our particular solution, which gives us:

y_0 = y(0) = 6e^(0) - 4(0) + 1 = 6+1 = 7

Now, to solve for g(t), we'll need to take the derivative of y(t) as g(t) is defined in terms of both y and y'. So, taking the derivative yields:

y'(t) = -12e^(-2t) - 4

All that is left to do is plug this into our ODE, so we have:

g(t) = y'(t) + y(t) = -12e^(-2t) - 4 + 6e^(-2t) - 4t +1 = -6e^(-2t) - 4t - 3

And, we're done.

Answer 3

Thank You so much! I got the due date extended for the assignment so this got my grade on the hw from a 75 to a 90! thank you!