MATH 1502 Midterm: MATH 1502 GT Exam1 Solutions
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Solutions: consider the following linear di erential equation y 8y + 16y = 0 (a) find the general solution y(x). Solve characteristic equation: r2 8r + 16 = 0. The only root is r = 4, hence, the general solution is y(x) = (ax + b)e4x, where a, b are constants. (b) solve the initial value problem with the conditions y(0) = 1, y (0) = 6. The conditions applied to the general solutions give: y(0) = b = 1 and. 1 x dx = (cid:2) 2x 1 x(cid:3)b. As b 1 the above expression converges to 4. Use ratio test: ak+1 ak ((k + 1)! Since the series is positive and the ratio limit is < 1, the series converges abso- lutely. The series converges, since it is alternating and the absolute value of the term decreases and approaches 0. however, the series. 1 k n have the partial sum n sn =
