Differential Equations Answers

d^2y/dx^2+4dy/dx-4y=sin3x

Given the initial value problem    x 0 1 − 2x1 − 3x2 = 0 x 0 2 + x1 + x2 = 0 x1(0) = √ 3, x2(0) = 0 , find x2(t). (a) x2(t) = e t 2 sin √ 3 2 t (b) x2(t) = e t  cos 1 2 t + sin 1 2 t  (c) x2(t) = −2e t 2 sin √ 3 2 t (d) x2(t) = 2e t 2 cos √ 3 2 t + sin √ 3 2 t ! (e) x2(t) = 2e t 2 sin √ 3 2 t 

For which values of A , B , C and k is the function yp(x) = x 2 (A x2 + B x + C) a particular solution of the equation 16 y 000 + 9 y 00 = 1458 x k−1 (2x − 1) ? (a) A = 27 , B = 165 , C = −880 , k = 2 (b) A = 27 , B = −219 , C = 1168 , k = 2 (c) A = 27 , B = −219 , C = 1168 , k = 4 (d) A = 54 , B = −411 , C = 2192 , k = 3 (e) A = 54 , B = −411 , C = 2192 , k = 4 

Which of the following is the solution of the initial value problem y (5) + 4y (4) + 4y (3) = 0 , y(0)= 0 , y0 (0)= − 1 2 , y(2)(0)= 5 , y(3)(0)=−14 , y (4)(0)= 36 ? (a) y = −1 − t 4 + t 2 2 + e −2t (b) y = 1 4 − t + t 2 4 − 1 4 e −2t − te−2t (c) y = − 1 4 + t + 1 4 e −2t − te−2t (d) y = −1 + t 2 − t 2 4 + (1 + t 2 )e 2t (e) y = 1 2 + t − 1 2 e 2t + t 2 e 2t

Which of the following is the appropriate form for a particular solution of the differential equation y (4) + 3y 00 − 4y = 2x + cos2 x − cosh x ? (a) yp = (A + Bx)e x + (C + Dx) cos(2x) + (E + F x) sin(2x) (b) yp = (Ax + B)e −x + (C + Dx) cos(−x) + (E + F x) sin(−x) (c) yp = Ax + Bx2 + (C + Dx) cos x + (E + F x) sin(−x) + Ge2x (d) yp = Ax + Bx2 + (C + Dx) cos(2x) + (E + F x) sin(−2x) + Ge−x (e) yp = A + Bx + Cx sin(−2x) + Dx cos(2x) + Exex + F xe−x

Solve the partial differential equation (3D^² +10DD^1+ 3D ^1^2)z=e^x-y

Solve

(D^(2)-DD'-2D'^(2)+2D+2D')z=sin(2x+y)

Solve

(D^(2)+DD'-6D'^(2))z=y cosx

orthogonal trajectories of y' = - (x/y)