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?
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Reduce the following PDE to a set of three ODEs by the method of separation of
variables
Obtain all the first and second order partial derivatives of the function:
f ( x, y) = sin xy
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
Obtain the Fourier series expansion for the following periodic function which has a
period of π:
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
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