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Solve the ODEusing the method of trial solution y = erx like in the second year differential equations

course, and also using the Laplace transform method.   A simple ICI'EIIE example and an Engineering problem... 1. Solve the CIDE
y&quot;+y’-2y=ﬂ y[0}=y'(ﬂ}=1
using the method of trial solution 3; = em like in the second year differential equations
course, and also using the Laplace transform method.
2. {A} The Laplace transform of a periodic function ﬁt} with period T is given by H T
cssn=% where 1115):] erred: {1 — 2-33)? 5 Suppose f[t} is a periodic function with period 6 such that 15(5) = . Use this Ffs} to show that 1 - e4“ £{f{1}}= m {B} Let f[t]l he the function of part [A] 1whose Laplace transform is given above. Suppose
Yﬂs} is the Laplace transform of the solution to the initial value problem y'if}=1ﬂf(t} with y(ﬂ} =n Find an expression for Y[s]. (C) By writing
1-e-3s
1 +p 3, as the infinite series
1 - e-38
00
1 te-38
= [(-1)' (e-378 - e-3(j+1)8 )
j=0
find the solution of the initial value problem of part (B) in a form using the infinite
series. The following result may be helpful:
Clu(t - a)g(t - a)} = e-asCig(t) }
where a is a constant and u(t) is the Heaviside step function.
(D) A beam of length 6 units is clamped at the ends r = 0 and r = 6. Suppose we apply a
load of 162 units at r = 2. Then the deflection 2(r) satisfies the differential equation
d' z
dar4
= 162 6(x - 2)
where o is the Dirac Delta function. Use Laplace transforms to find the solution of this
differential equation when the boundary conditions are
=(0) = ='(0) = 0, 2&quot;(0) = 144, 2&quot;(0) = -120
Note that Co(x - 2)} = e-2. Some of the Laplace transforms given in part (C) may
be useful.

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