 # Advanced Math Problems

## Transcribed 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 dierentiable 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 dierential equations x 0 = 4x − 2y y 0 = 7x − 5y x(0) = 1, y(0) = 2.

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