The solution of the differential equation (dθ)/dt=-kθ-θ_0, where k is constant, is ……

Q100 / 150 · MHT CET 2019 (Shift 2) · Mathematics

The solution of the differential equation $\frac{d θ}{d t} = - k (θ - θ_{0})$ , where k is constant, is ……

We have differential equation

\frac{d θ}{d t} = - k (θ - θ_{0}),

where k is constant

\Rightarrow \frac{d θ}{d t} + k θ = k θ_{0}

Which is linear differential equation in form of

\frac{d y}{d x} + P y = Q

∴ I F =_{e} \int k d t = e^{k t}

Therefore, required solution,

(θ) (e^{k t}) = \int (e^{k t} \times k θ_{0}) d t

\Rightarrow θ^{e k t} = e^{k t} θ_{0} + a

\Rightarrow θ = θ_{0} + a e^{- k t}

Hence, the correct answer is option: 'A'.