MATH300 Midterm: MATH 300 UofA Midterm
Document Summary
Mathematics 300 - sample midterm: solve the following di erential equation for u e x d dx(cid:0)ex du dx(cid:1) = x u(0) = 0, u(a) = 0. 0 < x < a: given the function f (x) =( cos x. 2( + ) ( 6= ) ( 6= ) ( 6= ) sin 2 x. 2 + ekx(k sin x cos x) ekx(k cos x + sin x) k2 + 2 k2 + 2. Solutions: since d dx(cid:0)ex du dx(cid:1) = xex, integrating we get ex du dx. = z xex dx + c1 = [xex z ex dx] + c1 therefore ex du dx = xex + ex + c1, and so du dx = x + 1 + c1e x. 2 x2 + x c1e x + c2 and u(0) = 0 = c1 = c2, while u(a) = 0 = c1 = a2 2a.
