- 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 = ex whereas, example two a solution might be 1
To show ex is a solution of y” + 2y’ – 3y = 0 :
y = ex , y’ = ex , y” = ex
Subbing into y” + 2y’ – 3y = 0
ex + 2* ex – 3* ex = 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