Questions:
1. Would appreciate some help on the following type of question please
2. How do I express the following inhomogeneous system of first-order differential equations for x(t) and y(t) in matrix form?
? = -2x - y + 12t + 12,
?= 2x-5y-5
3. How do I express the corresponding homogeneous system of differential equations, also in matrix form?
4. How do I find the eigenvalues of the matrix of coefficients and an eigenvector corresponding to each eigenvalue. From this how would I write down the complementary function for the system of differential equations?
5. How would you calculate a particular integral for the inhomogeneous system, and then find the general solution?
6. How would I determine the particular solution of the initial-value problem with the initial conditions x(0) = 3 and y(0) = 2
I have another problem below but on a similar topic:
If an object moves in the plane in such a way that its Cartesian coordinates (x, y) at time t satisfy the following homogeneous system of second-order differential equations:
?= -2x - y,
?= 2x-5y (the ? here should have two dots on it but I cannot find the character)
How would I:
Express the system in matrix form?
Find the general solution of the system?
I think this system undergoes simple harmonic motion in a straight line in two distinct ways but why?
And for each such simple harmonic motion how do I determine the angular frequency and the vector giving the direction of motion?