Solve the following ordinary differential equations:
i) dy/dx = y- x/x- 4y
ii) (2yx2 + 4)dy/dx +( 2y2x-3) =0
iii) d2y/dx2 + 3(dy/dx) - 10y = 3x2
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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.
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.
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
Reduce the following PDE to a set of three ODEs by the method of separation of
variables
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?
Obtain all the first and second order partial derivatives of the function:
f ( x, y) = sin xy
Solve the following ODE using the power series method:
(x + 2)y′′ + xy′ − y = 0