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MATH 308 – 531 Homework 1 Spring 2016 Due Thursday, Jan. 27 at the beginning of class. 1. Determine which of the following equations is liear or non linear? If the equation is non linear, circle all the non linear terms. (a) (b) (c) (d) (e) d2 y dy 3 2 + 4 + 9y = 2 cos t dt dt dy t =3−y dt d2 y dy + y − t2 = 0 2 dt dt dy ety =1 dt dy √ d3 y t2 + y 3 = 5t dt dt 2. Determine whether the given function is a solution to the given equation. d2 y + y = t2 + 2 dt2 dy d2 y (b) y = 2e3t − e2t , 2 − y + 3y = −2e2t dt dt (a) y = sin t + t2 , 3. Determine for which values of m the function y(t) = tm is a solution to the equation d2 y dy 3t2 2 + 11t − 3y = 0. dt dt 4. (a) Show that the function y(t) = c1 et + c2 e−2t is a solution to the equation d2 y dy + − 2y = 0 for any choice of the constants c1 and c2 . dt2 dt (b) Determine c1 and c2 so that the initial conditions y(1) = 1, y ′ (1) = 0 are satisfied. 5. Solve the diﬀerentiate equation/initial value problem. 1 − x2 dy = dx y2 (b) y −1 dy + yecos x sin x dx = 0 (a) 2 (c) (x + xy 2 )dx + ex ydy = 0 √ dy (d) = 2 y + 1 cos x, y(π) = 0 dx √ (e) y dx + (1 + x)dy = 0, y(0) = 1 1