Question

The homogeneous solutions to an ODE are sin(2t) and cos(2t). Suppose that the forcing function is 1.5 cos(2t) what is an appropriate form of the general solution?

1. y(t)=Acos2t +Bsin2t + C t cos(2t+ᶲ) ,        (b) y(t)=Acos2t +Bsin2t + C cos2t + Dsin2t
2. y(t)=Acos2t +Bsin2t,                                      (d) y(t)=Acos2t +Bsin2t + C cos(2t+ᶲ)

What is the total number of linearly independent solutions that the following ODE must have?

y" +5y'+6xy=sinx

1. Two       (b) Four                (c) Three              (d) Five

Answer 1

1) Since the coefficient of sine and cosine term is 2 in homogeneous solution i.e sin(2*t) and cos(2*t) and the coefficient of cosine term in forcing function is 1.5*cos(2*t) is also 2. The general solution must be written as follows:

y(t)=A*cos(2t)+B*sin(2t)+ct*sin(2*t+). Ans is a)

2) Total no of linearly independent solutions to an ODE=order of that ODE. Order of the ODE is 2. Since y" exists(second derivative exists) and hence Total no of linearly independent solutions to this ODE=2.

Ans is a)