Reference no: EM132911299

Problem. In this problem, answer whether the following statements are true or false. You need to justify your answer to get credits.

(a) For any constant matrix A, x'(t) = Ax(t) has a unique fixed point at the origin.

(b) Suppose we have two linearly independent solutions x₁ and x₂ of a 2-D 1st order linear system of ODEs. Then, two solutions x₁ + x₂ and x₁ - x₂ are also linearly independent. [Hint]: det (AB) = det (A) det (B).

(c) When we have repeated eigenvalues in solving x'(t) = Ax(t) with constant matrix A, solving the associated eigenvectors (not the generalized eigenvectors of rank k > 2) is never enough for finding the general solution.

(d) Denote X(t) as the fundamental matrix of x'(t) = A(t)x(t). The particular solution of the n-D inhomogeneous system of ODEs, x'(t) = A(t)x(t) + b(t), is then given by
x_p(t) = X(t) ∫ X^-1(t)b(t)dt. (1)

Denote a_k(t) as the k-th component of the vector X^-1(t)b(t). X^-1(t)b(t)dt is actually a n-by-1 vector with n indefinite integrals a_k(t)dt, k = 1, 2, ..., n, as it's components. Each of this indefinite integral can be expressed up to an arbitrary integration constant:
∫ a_k(t)dt = h_k(t) + c_k
where h_k(t) is a chosen anti-derivative of a_k(t). Regardless of what values of c_k ∈ R, k = 1, 2, ..., n, one chose for computing ak(t)dt, Eq. (1) always gives us a particular solution of x'(t) = A(t)x(t) + b(t).

(e) Suppose we are solving a 2nd order 1-D linear inhomogeneous ODE, x''(t) + ax'(t) + bx(t) = g(t) (2) where a and b are constants. After getting two linearly-independent particular solutions x₁(t) and x₂(t), we can always use the Abel's formula to get the Wronskian W (t) = e^-at
and then use

x_p = -x₁(t) ∫ g(t)x₂(t)/W(t) dt + x₂(t) ∫ g(t)x₁(t)/W(t) dt
to get the particular solution of Eq. (2).