1. x(t) = 2e2t + 4e−t cos(3t) − 2e−t sin(3t), y(t) = 5e2t + e−t cos(3t) + 7e−t sin(3t). " cos(t) − sin(t) # 2. ~x(t) = t 3. y(x) = 2 + 3x − x2 − 56 x3 + . . . dx 4. dt 5. = (x − 1)(x − 2)(x − 3), limt→∞ x(t) = 2 u(x, t) = − 12 γx2 + 6. (a) y(t) = 12 4t 7 e 1 2 − α + β x+α+2 + 97 e−3t − 2e2t . (b) P∞ an 1 < x(0) < 3. n γ n=1 n3 π 3 [(−1)n − 1] + 2 nπ y(t) = c1 t−2 + c2 t−2 ln(t). 7. (a) ut = 1 4 uxx , 0 < x < π, t > 0, u(0, t) = 10, t > 0, ux (π, t) = 0, t > 0, u(x, 0) = sin(3x/2) + 10, 0 < x < π. (b) u(x, t) = e−9t/16 sin( 3x 2 ) + 10. o 2 π2 t [α − (−1)n β] e−n sin(nπx)