Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Solutions to Review for Exam III MATH 2306 (Ritter) Sections Covered: 8, 9 This review is provided as a courtesy to give some idea of what material is covered. Nothing else is intended or implied. (1) Find the general solution of the homogeneous equation. (a) y 00 −2y 0 +5y = 0 y = c1 ex cos(2x)+c2 ex sin(2x) (b) y 00 +6y 0 +9y = 0 y = c2 e−3x +c2 xe−3x (c) y 00 −36y = 0 y = c1 e6x +c2 e−6x (d) y (4) +3y 00 −4y = 0 y = c1 cos(2x)+c2 sin(2x)+c3 ex +c4 e−x (e) y 000 +2y 00 +y 0 = 0 y = c1 +c2 e−x +c3 xe−x (f) 2y 00 −3y 0 −2y = 0 y = c1 e−x/2 +c2 e2x (2) Solve each IVP (a) y 00 −3y 0 +2y = 0 y(0) = 0, (b) y 00 +2y 0 = 0 y(1) = 0, y 0 (0) = 2 y 0 (1) = 1 (c) y 00 −2y 0 +5y = 0 y(0) = 0, y 0 (0) = 2 y = 2e2x −2ex 1 e2 y = − e−2x 2 2 y = ex sin(2x) (4) Find the general solution of each nonhomogeneous equation (a) y 00 +6y 0 +9y = ex +3e−3x (b) y 00 +y 0 −12y = 2x y = c1 e−3x +c2 xe−3x + 1 1 y = c1 e−4x +c2 e3x − x− 6 72 1 x 3 2 −3x e + xe 16 2 (c) y 00 +y = 4 cos x y = c1 cos x+c2 sin x+2x sin x (5) Determine the form of the particular solution. (Do not bother trying to find any of the coefficients A, B, etc.) (a) y 00 −4y 0 +5y = x cos 2x (b) y 00 +y = x3 +ex yp = (Ax+B) cos(2x)+(Cx+D) sin(2x) yp = Ax3 +Bx2 +Cx+D+Eex (c) y 00 −4y 0 +5y = xe2x sin x (d) y 00 −2y 0 +y = 1+ex yp = (Ax2 +Bx)e2x sin x+(Cx2 +Dx)e2x cos x yp = A+Bx2 ex (6) For each homogeneous equation, write out the characteristic equation. If the equation doesn’t have a characteristic equation, briefly state why. (a) 3 d3 y dy d4 y −2 + −4y = 0 dx4 dx3 dx (b) 4y 00 +2xy 0 +ex y = 0 3m4 −2m3 +m−4 = 0 none exists, it’s not constant coefficient (c) x3 y 000 +2x2 y 00 −4xy 0 +y = 0 (d) y (6) +16y (4) −12y 00 +y = 0 none exists, it’s not constant coefficient m6 +16m4 −12m2 +1 = 0 (7) For each of the following nonhomogeneous equations, determine whether the method of undetermined coefficients could be used to determine yp . If not, give a brief explanation. d4 y d3 y dy −2 + −4y = x3 ex yes, it could dx4 dx3 dx 1 (b) 4y 00 +2y 0 +y = nope, RHS is not of the correct type 1 + x2 (a) 3 (c) x3 y 000 +2x2 y 00 −4xy 0 +y = sin(2x)+x (d) y (6) +16y (4) −12y 00 +y = x ln x nope, LHS is not constant coefficient nope, RHS is not of the correct type