Differential Equations

• 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

x² – 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’ = 4x² + 3y

c) y’ = 6 x² + 6 y²