Practice problems Oleg Ivrii July 12, 2020 Oleg Ivrii Practice - - PowerPoint PPT Presentation

Practice problems

Oleg Ivrii July 12, 2020

Oleg Ivrii Practice problems

Exam topics

The exam will have 6 problems, the topics are:

1 Solve an IVP or a BVP 2 Gronwall’s inequality, continuation of solutions 3 ODEs with variable coefficients and Wronskians 4 Laplace transform 5 Boundary value problems and the maximum principle 6 Eigenvalue problems and the Sturm-Liouville Theorem Oleg Ivrii Practice problems

Solve an IVP or a BVP

• 1. Solve the initial value problem

y′′ + 3y′ + 2y = x3 with the initial conditions y(0) = y′(0) = 0.

Oleg Ivrii Practice problems

Solve an IVP or a BVP

• 2. Solve the boundary value problem

y′′ + y = sin(2x), y(0) = 0, y(π) = 0.

Oleg Ivrii Practice problems

Solve an IVP or a BVP

• 3. Show that the boundary value problem

y′′ + y = x, y(0) + y′(0) = 0, y(π/2) − y′(π/2) = π/2 has no solution.

Oleg Ivrii Practice problems

Solve an IVP or a BVP

• 4. Solve the following boundary value problem

y′′ + 2y′ + 10y = 0, y(0) = y(π/6), y(π/6) = y′(π/6).

Oleg Ivrii Practice problems

Gronwall’s inequality, continuation of solutions

• 1. Let y(x) be a solution of the IVP

y′ = y − y2, y(0) = a, 0 < a < 1. Show that a < y(x) ≤ 1 for all x ∈ (0, ∞).

Oleg Ivrii Practice problems

Gronwall’s inequality, continuation of solutions

• 2. (Exercise 3, Problem 1) Let y(x) be the solution of the initial

value problem

• y′(x) = y2 − x,

y(0) = 1, defined for x ∈ (−ε, ε). Show that y(x) extends to the interval [0, 1) with the bounds 1 + x < y(x) < 1 1 − x , x ∈ [0, 1).

Oleg Ivrii Practice problems

Gronwall’s inequality, continuation of solutions

• 3. (a) Show that the Riccati equation

y′ = 1 + x2 + y2 has no solution on the interval (0, π). (b) Show that the Riccati equation y′ = 1 + y2 − x2 has a unique solution defined on the entire real axis.

Oleg Ivrii Practice problems

Gronwall’s inequality, continuation of solutions

• 4. Show that the IVP

y′ = f (x, y), y(0) = a has at most one solution provided y → f (x, y) is non-decreasing for any x ∈ R.

Oleg Ivrii Practice problems

ODEs with variable coefficients and Wronskians

• 1. (a) Suppose u, v be two linearly independent solutions of the

second-order linear homogenous ODE y′′ + p(x)y′ + q(x) = 0. Show that p(x) = −W ′(u, v)(x) W (u, v)(x) , q(x) = W (u′, v′)(x) W (u, v)(x) . (b) Construct a second-order linear homogenous ODE which has solutions u(x) = x and v(x) = sin x.

Oleg Ivrii Practice problems

ODEs with variable coefficients and Wronskians

• 2. (Exercise 5, Problem 3) Consider the inhomogeneous Euler

equation y′′ − 2 x · y′ + 2 x2 · y = f (x). (a) Check that y1(x) = x solves the homogeneous Euler equation with f (x) = 0. Find a complementary solution y2(x). (b) Use the method of variation of parameters to write down the general solution of the inhomogeneous Euler equation.

Oleg Ivrii Practice problems

ODEs with variable coefficients and Wronskians

• 3. Find the general solution of

y′′′ − 2y′′ − y′ + 2y = 12e2x ex + 1

• Hint. All roots are integers or complex integers.

Oleg Ivrii Practice problems

ODEs with variable coefficients and Wronskians

• 4. (a) Find the general solution of

y′′′ + y′′ + y′ + y = ex. (b) Find the general solution of y′′′ + y′′ + y′ + y = tan x.

• Hint. All roots are integers or complex integers.

Oleg Ivrii Practice problems

Boundary value problems and the maximum principle

• 1. Show that the following boundary value problems have at least
• ne solution:

(a) y′′ = 1 + x2e−y, y(0) = 1, y(1) = 7, (b) y′′ = sin x cos y + ex, y(0) = 0, y(1) = 1.

Oleg Ivrii Practice problems

Boundary value problems and the maximum principle

• 2. Show that the following boundary value problems have at most
• ne solution:

(a) y′′ = y3 + x, y(0) = 0, y(1) = 1, (b) y′′ = y + cos y + x2, y(0) = 1, y(1) = 5.

Oleg Ivrii Practice problems

Boundary value problems and the maximum principle

• 3. (Exercise 5, Problem 4) Show that the boundary value problem

y′′ − xy = 0, x ∈ (0, 1), y(0) = 0, y(1) = 1 has a unique solution. Prove the bounds: x + x2 2 ≤ y(x) ≤ x, x ∈ [0, 1].

• Note. The problem asks you to prove three things: existence,

uniqueness and the relevant bounds. Each part requires some imagination but can be done separately.

Oleg Ivrii Practice problems

Boundary value problems and the maximum principle

• 4. (Exercise 11, Problem 4) Show that the initial value problem

y′′ + y′/x − y = 0, x ∈ (0, 1), y(0) = 1, y′(0) = 0 has a unique solution. Prove the bounds: 1 + x2/4 ≤ y(x) ≤ 1 + x2/3, x ∈ [0, 1].

• Note. The problem asks you to prove three things: existence,

uniqueness and the relevant bounds. Each part requires some imagination but can be done separately.

Oleg Ivrii Practice problems

Laplace transform

• 1. Solve the initial value problem

y′′ − 2y′ + 2y = 0, y(0) = 2, y′(0) = 0 with help of the Laplace transform.

Oleg Ivrii Practice problems

Laplace transform

• 2. Solve the initial value problem

y′′ − 4y′ + 4y = 0, y(0) = 1, y′(0) = 1 with help of the Laplace transform.

Oleg Ivrii Practice problems

Laplace transform

• 3. Solve the initial value problem

y′′ − 3y′ + 2y = g(t), y(0) = 0, y′(0) = 0, with help of the Laplace transform, where g(t) =

• sin t,

0 ≤ t ≤ π, 0, t ≥ 0.

Oleg Ivrii Practice problems

Laplace transform

• 4. The Bessel function of order zero has the power series expansion

J0(t) =

∞

• n=0

(−1)nt2n 22n(n!)2 . Show that L

• J0(t)
• (s) =

1 √ s2 + 1 and L

• J0(

√ t)

• (s) = e−s/4

s .

Oleg Ivrii Practice problems

Eigenvalue problems and the Sturm-Liouville Theorem

• 1. You are given that the Sturm-Liouville problem

y′′ = −λy, y(0) = 0, y(1) = 0, has normalized eigenfunctions φn(x) = √ 2 sin(nπx), λn = n2π2. Use this information to solve y′′ = sin(2πx) − sin(3πx), y(0) = 0, y(1) = 0.

Oleg Ivrii Practice problems

Eigenvalue problems and the Sturm-Liouville Theorem

• 2. You are given that the Sturm-Liouville problem

y′′ = −λy, y(0) = 0, y(1) = 0, has normalized eigenfunctions φn(x) = √ 2 sin(nπx), λn = n2π2. Find a sequence of coefficients {an} such that y = anφn solves y′′ = x − x2, y(0) = 0, y(1) = 0.

Oleg Ivrii Practice problems

Eigenvalue problems and the Sturm-Liouville Theorem

• 3. Solve the eigenvalue problem

−y′′ = λy, x ∈ (0, 1), with the boundary conditions y(0) = y′(1) = 0. Write down the associated eigenfunctions.

Oleg Ivrii Practice problems

Eigenvalue problems and the Sturm-Liouville Theorem

• 4. Write down a particular solution to the wave equation on

x ∈ [0, 1] and t ∈ [0, ∞]:              utt = uxx, u(t, 0) = u(t, 1) = 0, u(0, x) = 2 sin(2πx) − 3 sin(3πx), ut(0, x) = 0.

Oleg Ivrii Practice problems

Thank you for your attention!

Oleg Ivrii Practice problems