Question

The solution of the differential equation `("d"theta)/"dt" = - "k"(theta - theta_0)` where k is constant, is ______.

Options

• θ = θ₀ + ae^-kt

• θ = θ₀ + ae^kt

• θ = 2θ₀ - ae^kt

• θ = 2θ₀ - ae^-kt

Answer 1

The solution of the differential equation `("d"theta)/"dt" = - "k"(theta - theta_0)` where k is constant, is θ = θ₀ + ae^-kt.

Explanation:

We have differential equation

`("d"theta)/"dt" = - "k"(theta - theta_0)`, where k is constant

`= ("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"`

`=> theta "e"^"kt" = "e"^"kt" theta_0 + "a"`

`=> theta = theta_0 + "ae"^(- "kt")`