1. Express each of the following physical statements in the form of differential equation.
(i) Radium decays at a rate proportional to the amount Q present.
(ii) The population P of a city increases at a rate proportional to the product of population and to the difference between 5,00,000 and the population.
(iii) For a certain substance, the rate of change of vapor pressure P with respect to temperature T is proportional to the vapor pressure and inversely proportional to the square of the temperature.
(iv) A saving amount pays 8% interest per year, compounded continuously. In addition, the income from another investment is credited to the amount continuously at the rate of ₹ 400 per year.
2. Assume that a spherical rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
1. (i) dQ/dt = kQ (ii) dP/dt = kP(500000 − P) (iii) dP/dT = kP/T2 (iv) dx/dt = 2x/25 +400
2. dr/dt = −k