1. Specialist Mathematics
Solving differential equations (DEs).

2. Solving differential equations
Verifying solutions.
First and second order when dy/dx is the function of x.
First order when dy/dx is the function of y.
By separation of variables.
Using CAS.
Applications with DEs.
Sketching slope fields and Euler’s method
for numerical solutions of DEs.

3. General solution versus particular solution.
Example: Find the general solution to
general solution
Particular solution requires boundary conditions.
Example: Solve given that the solution curve passes through (2,0).
particular solution

4. Classification of differential equations (DEs).
The order of equation
The degree of equation
The highest derivative order that appears in a DE is called the order of the equation.
The highest power of a DE gives the degree.
is a first order DE
is a second order DE
is a third order DE
Linear and non-linear DEs.

5. Definitions and Terms
A differential equation (diff. eq., DE) is an equation that involves x, y, and some derivatives of y.
These are called ordinary differential equations (ODEs) because y is a function of only x.

6. Verifying solutions.

7. Real life applications.
Population growth / decay, where rate of change of population is proportional to the population at any time.
Newton’s Law of cooling – rate of change of temperature is proportional to the excess of the temperature above its surroundings.
Salt solutions type questions.
Rates in and rates out in a container.
DEs with related rates.

8. Ex 1. Verify that the given function is a solution to the given DE.
Find dy/dx first
Substitue back into the given DE
Simpilfy
LHS=RHS, true so verified.

9. Ex 2. Verify that the given function is a solution to the given DE.
Notice that y = 0 is a solution to both DEs. This is called the trivial solution.

10. Ex 3. Find values for m that would make
y = emx a solution of the DE 2y + 7y – 4y = 0.

11. Practice Problem
Verify that is a solution to the DE

12. Question 6 p 372

13. Ex 9A p 372
Questions 1a, 2c, d, e, g, 2 c, g, 3, 4, 5, 7