Reference no: EM132911291

Problem 1. [Solving 2-D system] Consider the following system of ODEs
{x'(t) = -3x - y

{y'(t) = bx - y

with an undetermined parameter b ∈ R. All the general solutions asked below should be in terms of b.
(a) For what value(s) of b is(are) the eigenvalues real and distinct? Find the general solution in such case(s).
(b) For what value(s) of b is(are) the eigenvalues complex? Find the general solution in such case(s).
(c) For what value(s) of b is(are) the eigenvalues repeated? Find the general solution in such case(s).
(d) For what value(s) of b is(are) the origin an attracting (stable) fixed point?

Problem 2. Consider
x²y"(x) + xy'(x) - y(x) = 0, x > 0.

(a) This is a Cauchy-Euler equation. Find the general solution.

(b) Now, solve this problem with a power series centered at x₀ = 1, y(x) = Σ^∞_n=0 a_n (x - 1)ⁿ. Find the recurrence relation an satisfies.

(c) Taylor expand the two solutions you found in (a) at x₀ = 1. Check that the two series satisfy the recurrence relation you found in (b).

(d) In (c), one of the solutions is an infinite power series. What is the radius of convergence for that series?

Problem 3. [Gravitational Equilibrium] A spaceship with mass m moves along a line along two stationary planets with mass m₁ and m₂, which are separated by a fixed distance r. Let x denotes the distance of the spaceship from m₁. The spaceship wants to find a spot to rest. By Newton's second law and gravitational formula, it's position is governed by

x"(t) = F (x)/m = Gm₁/(x - r)² - Gm₂/x², 0 < x < r

where G is the gravitational constant. Let's denote Gm₁ = a and Gm₂ = b for notation simplicity. Denote v = x'(t).

(a) Find the x coordinate of the fixed point (x*, v*) such that the spaceship can rest there at zero velocity.

(b) Classify the fixed point (x*, v*). Is the fixed point stable or not?

[Hint]: It turns out not much calculation is needed. You will need to consider some function associated with F (x) but the sign of the function is enough to finish this problem. You won't need to know the value of that function.

Problem 4. 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).