Reference no: EM132344797

Calculus & Matrix Algebra Assignment -

Show all working and give answers exactly where possible.

Q1. Evaluate the following integrals, showing all working;

(a) ₀∫¹6/√(5-3x²) dx

(b) ₁∫²(x-1)/((x+1)(2x-1)) dx

Q2. Solve the following separable differential equations. In each case show your solution is correct by substitution (to get full marks you must do this).

(a) dy/dx = y²sinx, y(0) = 2

(b) 1/x dy/dx = e^{x^2}/y.

Q3. Solve the following linear initial-value problems. Show all working and substitute your solution back into the DE to show it is correct (to get full marks you must do this).

(a) 2y'(x) + xy = 2x, y(0) = 5

(b) y'(x) - 2y/x = x, y(1) = 2.

Q4. Solve the following equations for z. Show your working and sketch the points on the complex plane.

(a) z⁵ + 1 = 0.

(b) z = (2-i)²/(3+2i).

(c) z² - 2iz + 1 = 0.

Q5. If possible, find the general solutions of the following 2nd order ODEs by trying y = e^λx. If you can't, give a reason.

(a) y'' - 2y' + 9y = 0

(b) y'' - 36y = 0.

Q6. The movement of a container crane in the wind satisfies the equation y'' + 2y' + βy = 0. For what values of β does the crane exhibit decaying oscillations? Write the solution in that case for general β.