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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
dx2
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.
A series circuit contains a resistor with R = 40Ω, an inductor with L = 2 H, a capacitor with
0.0025 F, and a 12 − V battery. The initial charge is Q = 0.001 C and the initial current is
0. Using the method of undetermined coefficients, find the charge at time t A series circuit contains a resistor with R = 40Ω, an inductor with L = 2 H, a capacitor with
0.0025 F, and a 12 − V battery. The initial charge is Q = 0.001 C and the initial current is
0. Using the method of undetermined coefficients, find the charge at time t
(y−(cosx)^2)dx+cosxdy=0
Find the partial derivative of f(x,y,z)=x^2y+6z^3xy
xy'=6y
Solve
(D^2+DD'-6D'^2)z= ycosx
Solve, by using the method of variation of parameter, the following differential equation
(D^2-3D+2)y = 1/(1+e^-x)
cos^2y+sinxcosxcosy p=sinycos^2x using u=sinx and v=siny
y^2logy=xpylogx+x^2p^2 by using logy=v,logx=u find G.S,S.S
p^2(x^2-a^2)-2xyp+y^2+a^4=0 find G.S and S.S
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