MAT 127: Calculus C, Fall 2010Homework Assignment 4WebAssign Problem due before 9am, Tuesday, 10/05 (all sections)20% bonus for submissions before 9am, Friday, 10/01Written Assignment due before5:30pm Tuesday, 10/05 in Math 3-121 if enrolled in L015:20pm Tuesday, 10/05 in Library W4525 if enrolled in L022:20pm Tuesday, 10/05 in Library W4540 if enrolled in L03Note the earlier due dates (because of Midterm I); solutions will be available after 7pmon 10/05)Please read Notes on Second-Order Differential Equations (called DE Notes below) thoroughlybefore starting on the problem set.Written Assignment: DE Notes 1,7,12,16; Problem D (next page)Show your work; correct answers without explanation will receive no credit, unless noted otherwise.Please write your solutions legibly; the graders may disregard solutions that are not readily read-able. All solutions must be stapled (no paper clips) and have your name and lecture number in theupper-right corner of the first page.Problem DBy Problem B on HW2, the first-order differential equationy0− by = f (x), y = y(x), b = const,can be solved by multiplying both sides by e−bx. This equation then becomese−bxy0= e−bxf(x)and can be solved by integrating both sides. Note that b is the root of the associated linear equationr − b = 0. This approach has an analogue for second-order inhomogeneous linear equationsy00+ by0+ cy = f (x), y = y(x), b, c = const. (1)(a) If r1, r2are the two roots of the quadratic equation r2+br +c = 0 associated to (1), show thate(r1−r2)x(e−r1xy)00= e−r2x(y00+ by0+ cy). (2)By (2), equation (1) is equivalent toe(r1−r2)x(e−r1xy)00= e−r2xf(x), y = y(x), (3)which can be solved by integrating twice.(b) Find the general solution y = y(x) to the differential equationy00+ 5y0+ 4y = e−x.Hint 1: comparing this equation with equation (1) above, what are b, c, f (x), r1, and r2here?How does the sentence following part (a) apply in this case?Hint 2: choosing the order of the roots wisely could simplify the computation.(c) Find the general solution y = y(x) to the differential equationy00+ 4y = 4 cos 2x.Hint 1: see the two hints above.Hint 2: replacing cos 2x by e2ixand then taking the real part of the resulting general solutionwould simplify the computation. This real part is the general (real) solution to theabove equation because cos 2x is the real part of e2ixand all coefficients in the equa-tion are real.Note: If you ask someone at MLC/RTC to help you with this problem, do not just point themto part (b) or (c), but ask them to read the introduction at the beginning of the problem. Theymay not know how to help you right away because an approach to more general equations of thisform is introduced in MAT 303. The approach described above is simpler, but is applicable to anarrower set of