Download Matemaattiset apuneuvot II Autumn 2016 Homework 2 Return friday

Matemaattiset apuneuvot II
Autumn 2016
Homework 2
Return friday 11.11 by 12 to 2nd floor A-wing boxes. Discussed on 8.11 and final
discussion on 15.11.
Staple papers together, and write the number of the group and your
teaching assistant’s name on the paper
1. Solve the initial value problem
y 00 + 2y 0 − 8y = 0,
y(0) = 3 and y 0 (0) = −12
2. Solve the differential equation 2y 00 − 3y 0 − 2y = 1 with initial conditions y(0) =
0, y 0 (0) = 1. (Use a guess for the individual solution of the complete equation!)
3. A pointlike body moves as a function of time t along the trajectory r(t) =
ti + e−2t j Calculate the
a) velocity v and speed v
b) unit tangent vector T = v/v
c) acceleration a
d) The components of the acceleration parallel to the trajectory T · a and
perpendicular to it.
4. The angular momentum of a pointlike body (mass m) with respect to the
origin of the coordinate system is defined with the equation L = r × p, where
p = mv = mdr/dt. According to the Newton’s law the force is F = ma =
dp/dt. Show that dL/dt = r × F and deduce that when the body moves
under the influence of a central force (F ∝ r), the angular momentum L is
constant.
5.
a) An insect falls from the ceiling on a poisonous region on the floor, to
the coordinate point (1,2). The amount of the poison can be described
with the scalar function M (x, y) = x2 + y 2 + xy + 5x + y + 14. To which
direction the insect should move, or to which direction the amount of the
poison decreases fastest?
b) Search for the point where the amount of the poison is at minimum.
(note: on an extremal point the gradient of a scalar field is 0).
Extra problem: Complex numbers z1 , z2 , z3 obey |zi | = 1 and z1 + z2 + z3 = 0. Show
that
1
1
1
+ +
= 0 and z1 z2 + z2 z3 + z1 z3 = 0.
z1 z2 z3