Obtain a solution of the wave equation
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
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__________ ________ ____ ________ _______ ______ _______ __________ _____ ____ _________.
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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 springâ€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.
Reduce the following PDE to a set of three ODEs by the method of separation of
variables
Obtain a solution of the wave equation
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
Obtain the Fourier series expansion for the following periodic function which has a
period of π:
Show that the function
i) u( x, y) = tan −1(y / x) is a solution of the two-dimensional Laplace’s equation.
ii) u(x , t ) = e−6t cos 2x is a solution of the one-dimensional heat equation.
Solve the following ODE using the power series method:
(x + 2)y′′ + xy′ − y = 0
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?