The function y_p(t) =ln(4 + 2 t), t > -4/2, is a particular solution to the differential equation y''+8y=g(t). Find g(t).
To find g(t), we need to first find the second derivative of y_p(t) and then substitute it into the given differential equation.
The given function is y_p(t) = ln(4 + 2t). Taking the first derivative, we have:
y'_p(t) = 2/(4 + 2t)
Now, taking the second derivative of y_p(t), we differentiate y'_p(t) with respect to t:
y''_p(t) = -4/(4 + 2t)^2
Now let's substitute y_p(t) and y''_p(t) into the given differential equation:
y'' + 8y = g(t)
Therefore, substituting the second derivative y''_p(t) into the equation, we get:
-4/(4 + 2t)^2 + 8ln(4 + 2t) = g(t)
So, g(t) is given by -4/(4 + 2t)^2 + 8ln(4 + 2t).