ASSIGNMENT OBJECTIVES

In this assignment, for the Pass criteria, you will need to show that you can:

L04.1 Analyse problems using first order differential uations

LO4 2 Solve first order differential equations using a range of techniques

LO4 3 Understand how second order differential equations arise in engineering contexts

L04.4 Solve second order differential equations

L04.5 Use first and second order differential equations in engineering situations

For the Merit criteria, you will need to achieve the Pass criteria above and show:

Task 1

You must answer both parts (a) and (b) Your work colleagues; who recently left the company have created a number of differential equations after running some simulations. Your task is to help recreate the original equation in that describes the relationship of y with respect to x, that these are derived from.

(a) L04.2

(b) L04.1

Solve the following differential equation by direct integration:

(i) dy/dx = 4 + 3x² dx

Solve the following by separation of variables:

(ii) dy/dx = 2y² (iii) dy/dx = 3x²e^-y (iv)dy/dx = 1/2y

(v)dy/dx = 2x/y (vi) dy/dx = 2x(y +1)

(b) After analysis of rock samples a scientific experiment yields a relationship modelled by the following equation:

dy/dx + 2y = e^-x

Solve this equation using an appropriate method with which you are familiar to find an expression for y in the same way.

Task 2

(a) Using Internet, books, or by drawing upon learning from other units, explain (with a short example) how a second order differential equation can be used as a mathematical model in an engineering context.

(b) As a result of some research on pressure differentials on certain strata of rock material, it is required to solve the differential equation below.

Given that that y=2 and dY/dx = 4 at x=0 solve the following differential equation using an appropriate method.

d²y/dx² = 3