QuestionQuestion

Transcribed TextTranscribed Text

1. Determine the Green's function and the general solution for the problem y 000(x) = f(x) y(0) = y”(0) = y 0 (1) = 0 2. Let a and B be constants. Under what conditions does the general first order boundary value problem y 0 (x) + f(x)y(x) = g(x), Ay(α) + By(β) = 0 have a unique solution for all g? When is this case, write down a formula for the solution. 3. Consider the problem of determining a solution to the initial value problem y 0 (x) = f(x, y), y(x0) = a where f is continuous dierentiable function. The problem is equivalent to the integral equation, labeled as (1) y(x) = a + Z x x0 f(ξ, y(ξ))dξ, Begins with an initial approximation φ(x) to the solution of (1) we can generate a sequence of successive approximations, φa, φa, φ3, . . . via φn+1(x) = a + Z x x0 f(ξ, φn(ξ))dξ, n = 0, 1, 2, . . . Follow the same procedure as described above to nd the sequence of solutions φn+1(x) for the initial value problem y 0 (x) = 2x(1 + y), y(0) = 0. 4. a) Find the Neumann series of (1) y(x) = x + λ Z x 0 (x − ξ)y(ξ)dξ b) Does the integral equation y(x) = sin(x) + 3 Z p 0 i(x + ξ)y(ξ)dξ have a solution? 5. Let B be a Banach space with norm || · ||. Let T : B → B (a) Show that if T is a contraction mapping, then it is continuous. (b) Let [a, b] ⊆ R, let K(t, s) be continuous on [a, b] × [a, b] with |K(t, s)| ≤ M for some positive constant M. Let f : [a, b] × R− > R be continuous and satisfy Lipschitz condition with Lipschitz constant L, for some constant L > 0. Also, p : [a, b] → R be continuous. Define T : C[a, b] → C[a, b] by (T x)(t) = p(t) + Z b a K(t, s)f(s, x(s))dx. 1 Show that if ML(b − a) ≤ α, for some real number α with α ∈ (0, 1), then T is a contraction on C[a, b]. (c) Let T : [0, 1] → [0, 1]. Show that if there is a real number α with α ∈ (0, 1) such that |T 0 (x)| ≤ α, for all x ∈ [0, 1] where T 0 is the derivative of T, then T is a contraction on [0, 1]. 6. Determine a change of variables which reduces the quadratic forms Q1 = 5x 2 1 + 4x1x2 + 3x 2 2 , Q2 = x 2 1 + 2x 2 2 simultaneously to canonical forms. 7. Let A = 4 −2 7 −5 (a) Calculate A100. Do not simplify your answer. (b) Solve the system of dierential equations x 0 = 4x − 2y y 0 = 7x − 5y x(0) = 1, y(0) = 2.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Advanced Math Problems
    $45.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Mathematics - Other Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats