MATH 2Z03 Midterm: sample_test_2_solutions_rebuilt.pdf
3 Dec 2014
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Use the variation of parameters method to nd the general solution of the di erential equation. 4 y 4 y + y = The auxilliary equation is written as 4 m2 4 m + 1 = 0 or 4 (m 1. 2 is double root of the auxilliary equation and the complementary solution has the form yc(x) = c1 ex/2 + x c2 ex/2, where c1 and c2 are arbitrary constants. In particular,we can choose the functions y1(x) = ex/2 and y2(x) = x ex/2 to form a fundamental system of solutions for the associated homogeneous equation. In standard form, the rhs of the de becomes f (x) = ex/2. = (cid:20)z (1 + x) ex dx(cid:21) ex/2 +(cid:20)z ex/2 ex/2 (1 + x) ex dx(cid:21) x ex/2 dx(cid:21) x ex/2 dx(cid:21) ex/2 +(cid:20)z. The wronskian associated with y1 and y2 is (cid:175)(cid:175)(cid:175)(cid:175)y1 y2 y0. 2 (cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175)ex x + 1 (cid:182) ex.
