Practice 1 - Practice Problems For Midterm 1 1 Find all the linear differential equations a t2 y ty et y = sin t b(1 y)y t2 y cos t = 0 c y sin y

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Unformatted text preview: Practice Problems For Midterm 1 1. Find all the linear differential equations: a. t2 y + ty + et y = sin t b. (1 + y )y + t2 y + cos t = 0 c. y + sin y = t d. t2 y + 5y = 7 2. Solve the following differential equations: x2 a. y = y(1+x3 ) b. y + y 2 sin x = 0 dy ex c. dx = y+ey 3. Solve the following initial value problems: a. y + 2y = te−2t , y (1) = 0 b. ty + 2y = et , y (1) = 1 c. (9x2 + y − 1)dx − (4y − x)dy = 0, y (1) = 0 d. 3x2 − 2y 2 + (1 − 4xy )y = 0, y (0) = 1 e. 2y + 3y − 2y = 0, y (0) = 1, y (0) = −9/2 f. 6y − 5y + y = 0, y (0) = 4, y (0) = 0 4. In each problem determine the critical (equilibrium) points, and classify each one as asymptotically stable, unstable, or semistable. a. dy = ay + by 2 , a > 0, b > 0, 0 ≥ 0 dt b. dy = ey − 1, −∞ < y0 < ∞ dt c. dy = y 2 (y 2 − 1), −∞ < y0 < ∞ dt Answers: 1. a, d 2. a. 3y 2 − 2 ln |1 + x3 | = c, x = −1, y = 0 b. y −1 + cos x = c, if y = 0; also y = 0 c. y 2 /2 + ey = ex + c 3. a. y = (t2 − 1)e−2t /2 t t b. y = et − e2 + t1 2 t 3 2 c. y = x−(24x +x 4−8x−16) d. x3 − 2xy 2 + y = 1 t e. y = −e 2 + 2e−2t f. y = 12et/3 − 8t/2 1/2 4. a. y = 0 is unstable b. y = 0 unstable c. y = −1 is asymptotically stable, y = 0 is semistable, y = 1 is unstable 1 ...
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This note was uploaded on 12/08/2011 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas at Austin.

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