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Find the eigenvalues and eigenfunctions of

𝑦

′′ +

𝑦 = 0 , 0 < 𝑥 < 3

𝑦′(0) = 0 , 𝑦′(3) = 0


An object which is released from a hot-air balloon at a height experiences a force of air resistance which is directly proportional to the velocity, v , after t seconds satisfies the differential equation dv/dt + 0.25v = 10, where v is in meter per second and t is seconds. Assuming that the initial velocity of the object is 0, find v in terms of t.
Q\Determine the following ODEs are exact or not? Why?
((y)dx+(x)dy/(x+y)^2)+dy=0
Q\Determine the following ODEs are exact or not? Why?
(y+cosx)dx+(x+siny)dy=0
Q\Determine the following ODEs are exact or not? Why?
(x/2)(cosy)y´=-1
Q\Determine the following ODEs are exact or not? Why?
(y^3)dx+3x^3y^2dy=0
Q\Determine the following ODEs are exact or not? Why?
(2xy)dx+2y^2dy=0
(2y^2-4x+5)dx+(4-2y+4xy)dy=0
For exactness, and solve it if it is exact.

Solve the ODE

y00 + 4y0 + 4y =

e􀀀2x

x3

using the method of variation of parameters


c) When a flexible cable of uniform density is suspended between two fixed points and hangs
of its own weight, the shape y = f(x) of the cable must satisfy a differential equation
d
2y
dx2
= k
s
1 + 
dy
dx2
where k is a positive constant. Consider the cable shown in the Figure 1 below.
Figure 1: Cable hanging between two points.
i) Let z =
dy
dx in the differential equation. Solve the resulting first-order differential equa-
tion (in z), and then integrate to find y. [6]
ii) Determine the length of the cable.
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