Hi! I'd be happy to help you with this differential equation problem. Based on the given information, we have the following:
1. dx/dt = 5(dax) + 25x = 0
2. x(0) = 5
3. Find d^2x/dt^2 when t = 7/4
First, let's rewrite the given differential equation:
dx/dt = -25x/5
Now, separate variables and integrate:
∫(dx/x) = -∫(5 dt)
ln|x| = -5t + C1
Now, exponentiate both sides:
x(t) = e^(-5t + C1)
To find the constant C1, use the initial condition x(0) = 5:
5 = e^(0 + C1)
C1 = ln(5)
So, the solution for x(t) is:
x(t) = e^(-5t + ln(5))
Now we need to find the second derivative, d^2x/dt^2:
dx/dt = -5e^(-5t + ln(5))
d^2x/dt^2 = 25e^(-5t + ln(5))
Finally, find d^2x/dt^2 when t = 7/4:
d^2x/dt^2 (7/4) = 25e^(-5(7/4) + ln(5))
This is the second derivative of x(t) at t = 7/4.