Reference no: EM132402929

Question 1 - f(x, y) = -2 + x(x-10) + y², g(x, y) = x(1+y²) + 7(1-y).

If (x₀, y₀) = (0, 1) is an initial estimate of the solution of f(x, y) = 0 and g(x, y) = 0, then using Newton's method find the next approximate solution.

Do all calculations to at least eight decimal digit accuracy. Your answers can be rounded to five decimal digit accuracy when entered. Or they can be entered as an exact rational expression.

For example: x₁ = 1.142605634, or 1.14261, or 649/568.

Question 2 - Consider the initial value problem: y' = 7x² + 5 y/x, y(1) = 4.

Using Euler's method: y_n+1 = y_n + hy'_n', x_n+1 = x_n + h,

with step-size h = 0.5, obtain an approximate solution to the initial value problem at x = 2.

Maintain at least eight decimal digit accuracy throughout all calculations.

You may express your answer as a five decimal digit number; for example 6.27181.

YOU DO NOT HAVE TO ROUND YOUR FINAL ESTIMATE.

Question 3 - Consider the initial value problem: y' = (6x+9)/(y+4)², y(0) = 2.

Using one step of the following explicit third order Runge-Kutta scheme

k₁ = hf(x_n, y_n),

k₂ = hf(x_n + 1/3 h, y_n + 1/3 k₁),

k₃ = hf(x_n + 2/3 h, y_n + 2/3 k₂),

y_n+1 = y_n + ¼(k₁+3k₃),  

Obtain an approximate solution to the initial value problem at x = 0.3.

Maintain at least eight decimal digit accuracy throughout all your calculations.

You may express your answer as a single five decimal digit number, for example 17.18263.

YOU DO NOT HAVE TO ROUND YOUR FINAL ESTIMATE.

Question 4 - Consider the boundary value problem: with u(0) = 2 and u(3) = 3

Find functions g and φ₁ such that u = g + α₁φ₁ is a quadratic approximation that satisfies the boundary conditions.

Your answer should consist of two expressions, the first representing the term g and the second representing the term φ₁. Both should be expressed in terms of the independent variable x.

Your answers should be expressed as a function of x using the correct syntax.

Question 5 - An autonomous system of two first order differential equations can be written as:

du/dt = f(u, v), u(t₀) = u₀,

dv/dt = g(u, v), v(t₀) = v₀.

A third order explicit Runge-Kutta scheme for an autonomous system of two first order equations is

k₁ = hf(u_n, v_n), l₁ = hg(u_n, v_n),

k₂ = hf(u_n, ½k₁, v_n + ½l₁), l₂ = hg(u_n + ½k₁, v_n + ½l₁),   

k3 = hf(u_n + 2k₂ - k₁, v_n+2l₂ - l₁), l₃ = hg(u_n + 2k₂ - k₁, v_n+2l₂ - l₁),

u_n+1 = u_n + 1/6(k₁+4k₂+k₃), v_n+1 = v_n + 1/6(l₁+4l₂+l₃).

Consider the following second order differential equation,

d²y/dt² - 3 dy/dt - 11y² = 0.7, with y(1) = 0 and y'(1) = 3.

Use the Runge-Kutta scheme to find an approximate solution of the second order differential equation, at t = 1.2, if the step size h = 0.1.

Maintain at least eight decimal digit accuracy throughout all your calculations.

You may express your answer as a five decimal digit number; for example 16.17423.

YOU DO NOT HAVE TO ROUND YOUR FINAL ESTIMATE.