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Department of Mathematics and Philosophy of Engineering
MPZ4230-Engineering Mathematics II
Assignments NO 02
Due Date: 11 -05 -2016 Academic Year 2015/2016
1. (a) A Projectile of mass m = 0.11kg
shoot vertically upward with initial velocity
v(0) = 8ms−1 is slowed due to the force of gravity
F g = −mg and due to air
resistance F r = −kv2 , where g =
9.8ms−2 and k = 0.002kgm−1 The differential
equation for the velocity is given by: mv = −mg − kv2
.
i. Find the velocity after 0.1s, 0.2s using the Taylor
method of order four.
ii. The nearest tenth of a second, determine when the projectile
reaches itsmaximum height and begin falling. Use the Taylor’s
method of order four.
2. Use Euler’s method with h = 0.025 and the Runge-
Kutta fourth order method with
h = 0.1 for find y(0.1) the problem y = t
+ y, 0 ≤ t ≤ 2, y(0) = −1.
3. Given y = 2
ty + t2et, 1 ≤ t ≤ 2, y(1) = 0, with exact
solution y(t) = t2(et − e). Find
y(1.04) and y(0.95)
(a) by using Euler’s method,
(b) by using modified Euler’s method,
(c) by using Taylor method,
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(d) by using Runge- Kutta method.
Find exact value of y(1.04) and y(0.95) .
4. Using finite difference method, find the values
of u(x, y) satisfying the Laplace equation∂ 2u
∂x2 +
∂ 2u
∂y2 = −2(x2
+ y2
), 0 < x, y < 1 at the pivotal points
of a square region0 ≤ x ≤ 1, 0 ≤ y ≤ 1 with boundary values
u(x, 0) = 1, u(x, 1) = 2, 0 ≤ x ≤ 1,
u(0, y) = 1 + y2, u(1, y) = 0, 0 ≤ y ≤ 1.
5. Use the forward difference method to approximate the solution
to the partial differential
equation ut(x, t) − uxx(x, t) = 0 for 0 < x
