The solution of the differential equationn `(d theta)/(dt)=-k(theta-theta_(0))` where k is constant, is . . . .

Options

A. `theta = theta_(0) + ae^(-kt)`
B. `theta = theta_(0) + ar^(kt)`
C. `theta = 2theta_(0) - ae^(kt)`
D. `theta = 2theta_(0) - ae^(-kt)`

Answer

A

Explanation

We have differential equation
`(d theta)/(dt) = - k (theta - theta_(0))`, we where k is constant
`implies (d theta)/(dt) + k theta = k theta_(0)`
Which is linear differential equation in the form of
`(dy)/(dx) + Py = Q`
`:. IF = e^(int kdt) = e^(kt)`
Therefore required solution,
`(theta) (e^(kt)) = int (e^(kt) xx k theta_(0)) dt`
`implies theta e^(kt) theta_(0) + a`
`implies theta = theta_(0) + ae^(kt)`