Newton’s Law of Cooling states that the rate at which the object's temperature

decreases is directly proportional to the difference between the temperature of the

object and the ambient temperature. A cup of tea (temperature = 90°C) is placed in a

room whose temperature is 30°C. After five minutes, the temperature of the tea has

dropped to 70°C. After how much time would the temperature of the tea be 50°C?

Reduce the following PDE to a set of three ODEs by the method of separation of
variables

Obtain the Fourier series expansion for the following periodic function which has a
period of π:

Solve the following ODE using the power series method:

(x + 2)y′′ + xy′ − y = 0

Solve the following ordinary differential equations:

for 0 ≤ x ≤ π and t > 0, and the following boundary and initial conditions:

u(0, t ) = u(π, t ) = 0,

u( x,0) = x(π − x) and ∂u ( x,0) = 0.

∂t

A block of mass 1 kg is attached to a spring of force constant k = 25/4 Nm⁻¹ . It is

pulled 0.3 m from its equilibrium position and released from rest. This spring‐block apparatus is submerged in a viscous fluid medium which exerts a damping force of

– 4 v (where v is the instantaneous velocity of the block). Determine of the position x(t)

of the block at time t.