Obtain all the first and second order partial derivatives of the function:
f ( x, y) = sin xy
Let z = f (x,
_____ ________.
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Obtain all the first and second order partial derivatives of the function:
f ( x, y) = sin xy
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
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
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
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.