LINEAR ALGEBRA & ORDINARY DIFFERENTIAL EQs
Assignment # 2
Question:1 Solve the following differential equations by using operator method:
(i) (D2+ 4)y= 4 sin2x(7)
(ii) (D4+D2)y= 3x2+ 6 sin x−2 cos x(8)
Question:2 Write the general form of the P.I. (without evaluating U.C.) for the following
ODEs and also indicate the value of k:
(i) y00 + 2y0+ 5y= 6 sin 2x+ 7 cos 2x(3)
(ii) y00 +y= 12 cos2x(3)
(iii) y000 +y00 + 3y0−5y= 5 sin 2x+ 10x2+ 7 (3)
(iv) y00 + 3y0+ 2y=ex(x2+ 1) sin 2x+ 4ex+ 3e−xcos x(3)
Question:3
(a) Solve by the method of U.C y00 + 2y0+y=excos x.(5)
(b) Solve y00 + 2y0+y=excos xby the method of variation of parameters . (5)
Question:4 Solve by the Cauchy-Euler method:
(i) 4x2y00 + 4xy0+ 3y= sin ln(−x), x < 0. (5)
(ii) (3x+ 4)2y00 + 10(3x+ 4)y0+ 9y= 0.(5)
Question:5 Solve the following D.E
d2y
dt2−3dy
dt −4y= 12e−3tsin 2t y(0) = 1 = y0(0)
(i) without using Laplace Transform. (10)
(ii) By using Laplace Transform. (10)
Question:6
(a) Consider the following functions:
f(t) = t2if 0 ≤t < 1
t3if t≥1g(t) = t2−t2U(t−1) + t3U(t−1)
(i) Show that f(t)≡g(t).
(ii) Find Laplace Transform of f(t) and g(t).
(3+3)
(b) Write the following functions in terms of Unit Step Function and also find
Laplace Transform.
(i)f(t) = t2if 0 ≤t < 1
t3if t≥1(ii)f(t) =
1 if 0 ≤t < 4
0 if 4 ≤t < 5
−1 if t≥5
(iii)see F ig (1)
(12)
Question:7
(a) The differential eq. for the instantaneous charge q(t) on the capacitor in an
R-L-C series circuit is given by
Ld2q
dt2+Rdq
dt +C−1q=E(t)q(0) = 0 = i(0).(1)
(i) Find the Laplace Transform of (1) if E(t) is given by Fig (2)
(ii) Solve (1) for any one of E(t) given.
(10)
(b) Solve y00 +y=δ(t−π
2) + δ(t−3π
2) with y(0) = 1, y0(0) = 0. (5)
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