Main objective of the assessment: The objective of this task is to solve a range of problems involving the numerical solution of differential equations. Solutions must be written up using LaTeX, and also numerical methods must be coded using MATLAB. Description of the Assessment: Each student must submit a report (a single .pdf file), written using LaTeX (article style). There is no hard page limit, but it should be possible to answer all questions successfully without writing more than 10 pages. All MATLAB codes used to generate results in the report should also be submitted in a .zip file, and it should be clearly stated in your answer to each question which code(s) correspond(s) to that question. The report should be clearly titled, and should address the solution of the following problems (in each question

More details;

### 1. Solving Differential Equations (DEs)

A **differential equation** (or “DE”) contains **derivatives** or **differentials**.

Our task is to **solve** the differential equation. This will involve integration at some point, and we’ll (mostly) end up with an expression along the lines of “*y* = …”.

Recall from the Differential section in the Integration chapter, that a differential can be thought of as a **derivative** where displaystylefrac{{left.{d}{y}right.}}{{left.{d}{x}right.}}dxdy is actually not written in fraction form.

**Examples of Differentials**

*dx* (this means “an infinitely small change in *x*“)

displaystyle{d}thetadθ (this means “an infinitely small change in displaystylethetaθ“)

displaystyle{left.{d}{t}right.}dt (this means “an infinitely small change in *t*“)

## Examples of Differential Equations

### Example 1

We saw the following example in the Introduction to this chapter. It involves a derivative, displaystylefrac{{left.{d}{y}right.}}{{left.{d}{x}right.}}dxdy:

displaystylefrac{{{left.{d}{y}right.}}}{{{left.{d}{x}right.}}}={x}^{2}-{3}dxdy=x2−3

As we did before, we will integrate it. This will be a **general solution** (involving *K*, a constant of integration).

So we proceed as follows:

displaystyle{y}=int{left({x}^{2}-{3}right)}{left.{d}{x}right.}y=∫(x2−3)dx

and this gives

displaystyle{y}=frac{{x}^{3}}{{3}}-{3}{x}+{K}y=3×3−3x+K

But where did that *dy* go from the displaystylefrac{{{left.{d}{y}right.}}}{{{left.{d}{x}right.}}}dxdy? Why did it seem to disappear?