4. Find the general solution of the differential equation y” - 2y' + y = e^t ln(t), t > 0. (3) (

First, we find the general solution of the homogeneous equation y'' - 2y' + y = 0. This has the

Question

4. Find the general solution of the differential equation y'' - 2y' + y = e^t ln(t), t > 0. (3) (a) Find the general solution y_H(t) = c_1y_1(t) + c_2y_2(t) of the homogeneous equation y'' - 2y' + y = 0. (b) Use variation of parameters to find a particular solution of (3) in the form y_P(t) = u_1(t)y_1(t) + u_2(t)y_2(t). A good place to start is by solving for the unknown functions u_1'(t) and u_2'(t) in the matrix equation [[y_1, y_2], [y_1', y_2']] [u_1', u_2'] = [0, e^t ln(t)]. Then compute antiderivatives u_1(t) = integral u_1'(t) dt, u_2(t) = integral u_2'(t) dt. A particular solution of (3) is y_P(t) = u_1(t)y_1(t) + u_2(t)y_2(t). Simplify the particular solution as much as possible. (c) Find the general solution of (3), y(t) = y_H(t) + y_P(t).

          4. Find the general solution of the differential equation y'' - 2y' + y = e^t ln(t), t > 0. (3) (a) Find the general solution y_H(t) = c_1y_1(t) + c_2y_2(t) of the homogeneous equation y'' - 2y' + y = 0. (b) Use variation of parameters to find a particular solution of (3) in the form y_P(t) = u_1(t)y_1(t) + u_2(t)y_2(t). A good place to start is by solving for the unknown functions u_1'(t) and u_2'(t) in the matrix equation [[y_1, y_2], [y_1', y_2']] [u_1', u_2'] = [0, e^t ln(t)]. Then compute antiderivatives u_1(t) = integral u_1'(t) dt, u_2(t) = integral u_2'(t) dt. A particular solution of (3) is y_P(t) = u_1(t)y_1(t) + u_2(t)y_2(t). Simplify the particular solution as much as possible. (c) Find the general solution of (3), y(t) = y_H(t) + y_P(t).

4. Find the general solution of the differential equation y” - 2y' + y = e^t ln(t), t > 0. (3) (a) Find the general solution yH(t) = c1y1(t) + c2y2(t) of the homogeneous equation y” - 2y' + y = 0. (b) Use variation of parameters to find a particular solution of (3) in the form yP(t) = u1(t)y1(t) + u2(t)y2(t). A good place to start is by solving for the unknown functions u1'(t) and u2'(t) in the matrix equation [[y1, y2], [y1', y2']] [u1', u2'] = [0, e^t ln(t)]. Then compute antiderivatives u1(t) = integral u1'(t) dt, u2(t) = integral u2'(t) dt. A particular solution of (3) is yP(t) = u1(t)y1(t) + u2(t)y2(t). Simplify the particular solution as much as possible. (c) Find the general solution of (3), y(t) = yH(t) + yP(t).

Added by Sarah B.

Question

Please give Ace some feedback