(2) (12 points each) Solve the following initial value problerns: (a) y\"-3-0, w

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(2) (12 points each) Solve the following initial value problerns: (a) y"-3-0, with v(O)2,(0) 4.( is a function of t). Reduction of Order: u and so C1 = 0. = 3y2 and so u-y3 +Cl . At t = 0, y = 2,0 4 0. y = 2 and so-V2-C2. y-V2--ya (1-1). This simplifies to y 2(1 -1) 2.] (b) (1 + e"(1 + xy)) dr + (rey + 2y) dy 0, with y(0) = 3. = So H'(x) = 1 and H(z) = x The general solution is zezy + y2 +z = C. When x = 0, y = 3 and so C = 9. (3) (9 points) Assurne that yl , y2 are solutions of the equation y" +py,+gy = 0 where p and q are functions of t. (a) Define the Wronskian W of vi,V2 (b) Show that the Wronskian satisfies a first order linear differential equation. That is, derive the equation. (c) Use the equation in (b) to show that either W is identically zero or it is never zero. So 0 = pW + W, with solution W. Cerp(-/p(t)dt). The exponential is always positive and so if C 0 then W is never zero.

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