Unformatted text preview: MEC702 Final Exam Revision Questions Section A 1. Solve the ODE. Find a general solution. For IVPs, solve completely.
(a) y3y0 + x3 = 0
(b) y0 = sec2 y
(c) y0 sin 2πx = πy cos 2πx
(d) xy0 = y2 + y, (Set xy = u)
(e) xy0 = x + y, (Set xy = u)
2. Test for Exactness. If exact, then solve. If not, then use an integrating
factor. If IVP, then solve completely.
(a) sin x cos ydx + cos x sin ydy = 0.
(b) e3θ (dr + 3rdθ) = 0.
(c) (x2 + y2)dx − 2xydy = 0.
(d) 3(y + 1)dx = 2xdy, (y + 1)x−4
3. Find the general solution for the linear ODEs. If IVP, then solve completely.
(a) y0 − y = 5.2
(b) y0 = 2y − 4x
(c) y0 + ky = e−kx
(d) y0 + 2y = 4 cos 2x, y( 41 π) = 3
(e) xy0 = 2y + x3ex 1 Section B 1. Find the Wronskian.
(a) e4x , e−1.5x .
(b) e−0.4x , e−2.6x .
(c) x3, x2
(d) x, x1
2. Solve the EulerCauchy Equations.
(a) x2y00 − 4xy0 + 6y = 0, y(1) = 0.4, y0(1) = 0.
(b) x2y00 + 3xy0 + 0.75y = 0, y(1) = 1, y0(1) = −1.5.
(c) x2y00 + xy0 + 9y = 0, y(1) = 0, y0(1) = 2.5.
(d) x2y00 + 3xy0 + y = 0.
3. Find a general solution y = yh + yp where yp is found using the Method
of Undetermined Coecients.
(a) y00 + 5y0 + 4y = 10e−3x .
(b) 10y00 + 50y0 + 57.6y = cos x.
(c) y00 + 3y0 + 2y = 12x2.
(d) 3y00 + 27y = 2 cos x + cos 3x. 2 Section C 1. Find a general solution y = yh + yp where yp is found using the Method
of Variation of Parameters.
(a) y00 + 9y = sec 3x.
(b) y00 + 9y = csc 3x.
(c) y000 − 2y00 − 9y0 + 18y = e2x
(d) yiv + 5y00 + 4y = 90 sin 4x
(e) yiv − 5y00 + 4y = 10e−3x
2. Solve the IVPs by the Laplace transform. Show all details.
(a) y0 + 5.2y = 1.4 sin 2t, y(0) = 0
(b) y0 + 2y = 0, y(0) = 1.5
(c) y00 − y0 − 6y = 0, y(0) = 11, y0(0) = 28
(d) y00 + 9y = 10e−t, y(0) = 0, y0(0) = 0
(e) y00 − 14 y = 0, y(0) = 12, y0(0) = 0 3 Section D 1. Determine the type of critical point and its stability as well.
a.) y10 = y1
y20 = 2y2
c.) y10 = y2
y20 = −9y1 b.) y10 = −4y1
y20 = −3y2 d.) y10 = 2y1 + y2
y20 = 5y1 − 2y2 2. Solve the following initial value problems.
a.) y10 = y1 + y2 ,
y20 y2 (0) = 6 y1 (0) = 7, y2 (0) = 7 y1 (0) = 7, y2 (0) = 2 = 4y1 + y2 b.) y10 = 3y1 + 2y2 ,
y20 y1 (0) = 1, = 2y1 + 3y2 c.) y10 = −y1 + 5y2 ,
y20 = −y1 + 3y2 3. Solve the given higherorder ODEs.
(a) y000 + 25y0 = 0.
(b) yiv + 2y00 + y = 0.
(c) yiv − 4y00 = 0.
(d) (D3 − D2 − D + 1)y = 0. 4 Section E 1. Determine if the given functions are odd or even or neither. Find the
Fourier series for the function. Show details of your work.
(a)
(b)
(c)
(d)
(e) (
−x, −1 < x < 0
p = 2L
f (x) = x =
+x, 0 < x < 1
(
1,
−2 < x < 0
p = 2L
f (x) =
−1, 0 < x < 2
(
x + L, −L < x < 0
p = 2L
f (x) =
L − x, 0 < x < L
(
0, −π < x < 0
p = 2L
f (x) =
1, 0 < x < π
(
0, −0.5 < x < 0
p = 2L
f (x) =
x, 0 < x < 0.5 , period . , period , period , period , period . . . 2. Find the type, transform to normal form and solve. Show your work in
details.
(a) uxx − 64uyy = 0.
Solution: u(x, y) = f1 (x + 8t) + f2(x − 8t)
(b) uxx − 16uyy = 0.
Solution: u(x, y) = f1 (x + 4t) + f2(x − 4t) 5 ...
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 Winter '16
 Alveen Chand
 Laplace, Cos, Periodic function, general solution, 2L

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