- Applications of Calculus, in particular, differential equations.

Differential equations

Differential equations differ from normal equations in that the solution is a function, or a set of functions.

For example a differential equation is

**y” + 2y’ – 3y = 0 **(example 1)

Whereas, an example of a normal equation is

**x ^{2} – 1 = 0 **(example 2)

In our first case, a potential solution is ** y = e ^{x}** whereas, example two a solution might be 1

To show **e ^{x}** is a solution of

**y” + 2y’ – 3y = 0**:

y = e^{x} , y’ = e^{x} , y” = e^{x}

Subbing into **y” + 2y’ – 3y = 0**

e^{x} + 2* e^{x} – 3* e^{x} = 0, therefore, true

Questions.

Question 1. [2 marks]

Show that y = e^{-3x } is a solution to y” + 2y’ – 3y = 0

Question 2. [3 marks, 1 mark each]

What is the order of the following equations

a)

b)

c)

Question 3. [6 marks, 2 marks each]

Draw the differential direction field maps for the following differential equations

a) y’ = 3x + 2y -3

b) y’ = 4x^{2} + 3y

c) y’ = 6 x^{2} + 6 y^{2}