MAT1001 (Calculus and Laplace Transform)
Assignment - 2
1. (a) Find equations for the tangent plane and normal line to the surface z =
x2 + y 2 at the point (2, −1, 5).
(b) Find the directional derivative of φ = 4e2x−y+z at the point (1, 1, −1) in a
direction toward the point (−3, 5, 6).
2. If A = (2xy + z 3 ) i + x2 j + 3xz 2 k
´
(a) Prove that C A · dr is independent of the curve C joining two given points
P1 (1, −2, 1) and P2 (3, 1, 4).
(b) Show that there is a differentiable function φ such that A = ∇φ and find it.
(c) Also find the work done by A acting on a moving object from P1 to P2 .
˜
3. Evaluate S A · ndS for A = 4xzi − y 2 j + yzk and S is the surface of the cube
bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0 and z = 1.
4. Verify Gauss divergence theorem for F = 4xi − 2y 2 j + z 2 k taken over the region
bounded by x2 + y 2 = 4, z = 0 and z = 3.
5. (a) Solve the following initial-value problem using method of undetermined coefficients:
dy
d2 y
− 2 + y = 2xe2x + 6ex , y(0) = 1, y 0 (0) = 0.
2
dx
dx
(b) For the following differential equations set up the correct form of linear combination of functions with undetermined literal coefficients to use in finding
a particular integral by the method of undetermined coefficients. (Do not
actually find the particular integral.)
dy
d2 y
+ 6 + 13y = xe−3x sin 2x + x2 e−2x sin 3x
2
dx
dx
6. Find the solution to the following differential equation on any interval not containing x = 0
dy
d2 y
x2 2 + x + 4y = 2x ln x.
dx
dx
7. Show that the differential equation for the current i in an electrical circuit containing an inductance L and a resistance R in series and acted on by an electromotive
force E sin ωt satisfies the equation Ldi/dt + Ri = E sin ωt. Find the value of the
current at any time t, if initially there is no current in the circuit.
√ cos t
√
.
8. Find L
t
1
−1
9. Using convolution theorem, find L
.
(s + 1) (s2 + 2s + 2)
10. Determine a solution for the given initial value problem y 00 + 4y 0 + 5y = δ(t − 3),
y(0) = 0, y 0 (0) = 0 using Laplace transform.
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