Systems of Linear Differential Equations - One Line Questions

1. Consider the system dy/dt = -y + 2z, dz/dt = -2y - z. The eigenvalues are lambda = -1 +/- 2i. What is the real part of the eigenvalues? — -1
2. Consider the system dy/dt = y + 2z, dz/dt = 3y + 2z. In matrix form, this is X' = AX, where X = [y, z]^T. What is the matrix A? — [[1, 2], [3, 2]]
3. If the eigenvalues of A are distinct real numbers lambda1, lambda2, ..., lambdan, then a fundamental set of solutions is given by: — {v1*e^(lambda1*t), v2*e^(lambda2*t), ..., vn*e^(lambdan*t)}
4. Consider the system dy/dt = z, dz/dt = -y. The coefficient matrix is A = [[0, 1], [-1, 0]]. The eigenvalues are: — +/- i
5. For the system dy/dt = y, dz/dt = 2z, the eigenvalues of the coefficient matrix are: — 1 and 2
6. Consider the system dy/dt = -y + 2z, dz/dt = -2y - z. The eigenvalues are lambda = -1 +/- 2i. What is the imaginary part of the eigenvalues? — 2
7. Consider the system dy/dt = 3y + z, dz/dt = y + 3z. The eigenvalues are lambda = 2 and lambda = 4. What is tr(A)? — 6
8. Consider the system dy/dt = 3y + z, dz/dt = y + 3z. The eigenvalues are lambda = 2 and lambda = 4. What is det(A)? — 8
9. In the matrix form X' = AX, what does X represent? — A column vector of the dependent variables
10. In the matrix form X' = AX, what does A represent? — A square matrix of coefficients, possibly functions of x
11. If v is an eigenvector of matrix A corresponding to eigenvalue lambda, then X(t) = v*e^(lambda*t) is a solution to X' = AX if: — A*v = lambda*v
12. A fundamental set of solutions for X' = AX consists of n linearly independent solutions for an n x n system. What is the general solution formed from these? — A linear combination of the fundamental solutions
13. What is a fundamental matrix for the system X' = AX? — A matrix whose columns form a fundamental set of solutions
14. What is an eigenvector corresponding to an eigenvalue lambda of a matrix A? — A non-zero vector v such that Av = lambda*v
15. What is a generalized eigenvector for a repeated eigenvalue lambda? — A non-zero vector w such that (A - lambda*I)w = v, where v is an eigenvector
16. What is the definition of a linear system of differential equations? — An equation where the dependent variables and their derivatives appear only to the first power and are not multiplied together
17. The method of variation of parameters can be used to find a particular solution X_p(t) for X' = AX + F(t) for: — Any continuous function F(t)
18. For the system dy/dt = z, dz/dt = -y, with eigenvalues +/- i, the origin is a: — Center
19. If all eigenvalues are purely imaginary (real part is zero), the origin is typically a: — Center (for linear systems)
20. The characteristic equation for a 2x2 matrix A = [[a, b], [c, d]] is: — det(A - lambda*I) = 0
21. If the eigenvalues of A are lambda1, ..., lambdan, the determinant of the fundamental matrix Phi(t) is given by: — det(Phi(t)) = det(Phi(0)) * e^(tr(A)*t)
22. For a system X' = AX with a single eigenvalue lambda of algebraic multiplicity n, if the geometric multiplicity is also n, then A is: — Diagonalizable
23. How is the Wronskian related to the coefficient matrix A for X' = AX? — dW/dt = tr(A) * W
24. What is the general form of a system of first-order linear differential equations? — dy/dx = a11*y + a12*z, dz/dx = a21*y + a22*z
25. If lambda = alpha +/- i*beta are complex eigenvalues with eigenvector v = a + i*b, the two real linearly independent solutions are: — e^(alpha*t) * ( (a*cos(beta*t) - b*sin(beta*t)) , (a*sin(beta*t) + b*cos(beta*t)) )
26. For a fundamental set of solutions {X1(t), X2(t), ..., Xn(t)} of X' = AX, the Wronskian W(t) = det([X1(t) | X2(t) | ... | Xn(t)]) is: — Either identically zero or never zero
27. For a system with complex conjugate eigenvalues lambda = alpha +/- i*beta, the real solutions can be derived from a single complex solution X(t) = v*e^((alpha + i*beta)*t) using: — Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)
28. If lambda is an eigenvalue of matrix A, then e^(lambda*t) is related to the solution of X' = AX by: — It is a fundamental solution component
29. If a system X' = AX has a solution X(t) = [e^(2t), 2e^(2t)]^T, what is the eigenvalue and eigenvector of A? — lambda = 2, v = [1, 2]^T
30. For a system X' = AX with constant coefficients, if A has eigenvalues lambda1 and lambda2, then det(A) is equal to: — lambda1 * lambda2
31. For a system X' = AX with constant coefficients, if A has eigenvalues lambda1 and lambda2, then tr(A) is equal to: — lambda1 + lambda2
32. The equation X' = AX is a system of linear differential equations: — Autonomous (or homogeneous if all coefficients are constants)
33. The method of undetermined coefficients can be used to find a particular solution X_p(t) for X' = AX + F(t) if F(t) has a specific form. What is this form? — Polynomials, exponentials, sines, cosines, or combinations thereof
34. The process of finding solutions for X' = AX with constant coefficients involves finding: — Roots of the characteristic polynomial of A
35. If eigenvalues have opposite signs (one positive, one negative real part), the origin is a: — Saddle point
36. The behavior of solutions to X' = AX near the origin (t -> infinity) is determined by the sign of the real parts of the eigenvalues. If all real parts are negative, the origin is a: — Stable node
37. If at least one eigenvalue has a positive real part, the origin is a: — Unstable node
38. What is the Wronskian of a set of solutions to a linear system? — The determinant of the matrix formed by the fundamental solutions
39. When solving X' = AX using eigenvalues and eigenvectors, if an eigenvalue lambda has geometric multiplicity less than its algebraic multiplicity, it indicates: — The need for generalized eigenvectors to form a fundamental set of solutions
40. What is the trace of a square matrix? — The sum of the diagonal elements
41. What happens if a 2x2 matrix A has a repeated eigenvalue lambda? — There might be only one linearly independent eigenvector
42. For the system dy/dt = y, dz/dt = 2z, the corresponding eigenvectors are: — v1 = [1, 0]^T, v2 = [0, 1]^T
43. A system of linear differential equations can be represented in matrix form as: — X' = AX
44. What is the general solution for X' = AX when A has distinct eigenvalues lambda1, lambda2 and corresponding eigenvectors v1, v2? — X(t) = c1*v1*e^(lambda1*t) + c2*v2*e^(lambda2*t)
45. For a system X' = AX with constant coefficients, what is the form of a typical solution? — X(t) = c1*v1*e^(lambda1*t) + c2*v2*e^(lambda2*t) + ...
46. If Phi(t) is a fundamental matrix for X' = AX, then the general solution is given by: — X(t) = Phi(t) * C, where C is a constant vector
47. The solution X(t) = v*e^(lambda*t) is derived from the assumption that the solution is of the form: — X(t) = v*e^(lambda*t)
48. For a non-homogeneous system X' = AX + F(t), what is the general solution? — X(t) = X_h(t) + X_p(t), where X_h is the homogeneous solution and X_p is a particular solution
49. If a repeated eigenvalue lambda has only one linearly independent eigenvector v, what form do the solutions take? — X1(t) = v*e^(lambda*t) and X2(t) = (v*t + w)*e^(lambda*t), where w is a generalized eigenvector
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