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DEfallHW9

# DEfallHW9 - MTH 2201 Diﬀerential Equations Homework 9...

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Unformatted text preview: MTH 2201 Diﬀerential Equations Homework 9 : Second Order Linear ODE Fall 2009 1. In the problems below, y1 (x) is a solution of the given DE. Use reduction of order method to ﬁnd the second solution of the DE. (a) y − 4y + 4y = 0; (b) y + 16y = 0; (c) 9y − 12y + 4y = 0; (d) x2 y − 7xy + 16y = 0; y1 = e2x . (Solution: y2 = xe2x ) y1 = e 3 . (Solution: y2 = xe 3 .) y1 = x4 . (Solution: y2 = x4 ln |x|.) y1 = x + 1. (Solution: 2x 2x y1 = cos 4x. (Solution: y2 = cos 4x) (e) (1 − 2x − x2 )y + 2(1 + x)y − 2y = 0; y2 = x2 + x + 2.) 2. Find the general solution of (i) 4y + y = 0. (ii) y + 8y + 16y = 0. (ii) 8y + 2y − y = 0. (iv ) y − y = 0 3. Consider y + y − 6y = 0. (i) Compute the solution φ satisfying φ(0) = 1, φ (0) = 0. (ii) Compute the solution ψ satisfying ψ (0) = 0, ψ (0) = 1. 4. Find all solutions φ of y + y = 0 satisfying φ(0) = 1, φ(π/2) = 2 5. Let φ be a solution of the equation y + a1 y + a2 y = 0, where a1 , a2 are constants. If ψ (t) = e(a1 /2)t φ(t). Show that ψ satisﬁes the DE y + ky = 0, where k is some constant. 6. Determine the values of α, for which all solutions of y − (2α − 1)y + α(α − 1)y = 0, tend to zero as t → 0. 7. Find the general solution of (i) y − 4y − 5y = 0 (iii) 16 (ii) d4 y d2 y + 24 2 + 9y = 0. dx4 dx dy d5 y − 16 =0 5 dx dx 1 ...
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