Question

For the system below: a. Find the general solution b. Perform the qualitative analysis. c. Extra points: Find the associated second order equation and solve it. Compare with the solution obtained i Justify your answers dx dt +3x+y dy dt 2x+2y

Answer 1

To find the general solution of the system, we write the system of equations in matrix form and find the eigenvalues and eigenvectors. For the qualitative analysis, we examine the stability of the system by analyzing the eigenvalues. As an extra point, we find the associated second-order equation and solve it to compare with the solution obtained.

Step-by-step explanation:

To find the general solution of the system:

We first write the system of equations in matrix form:

[dx/dt, dy/dy] = [1, 1][x, y] + [0, -1]

Then, we find the eigenvalues and eigenvectors of the coefficient matrix [1, 1][x, y] to obtain the general solution.

To perform the qualitative analysis:

We analyze the stability of the system by examining the eigenvalues of the coefficient matrix. If the eigenvalues have negative real parts, the system is stable.

For the extra points:

We find the associated second-order equation by substituting x = e^(rt) into the first equation and solving for r. Then we solve the second-order equation to obtain the solution and compare it with the solution obtained previously.