1. MTH1170 Differential Equations

2. Differential Equations
A differential equation is a mathematical equation that relates a function with its derivative. We will be solving first order differential equations using the technique of separation of variables.
For example: y’ = y - x  is a differential equation. This is is a differential equation because both y, and y prime are contained within the expression. 

3. First Order Differential Equation
The order of a differential equation is dictated by the highest order derivative contained within the equation. For example: y’ = y - x is a first order differential equation because the highest derivative of y is the first derivative. For example: y’’ = y - x is a second order differential equation because the highest derivative of y is the second derivative.

4. Solutions to Differential Equations
A differential equation will have solutions of the form: y = f(x) The solution will define the variable y as some function of x.

5. General Solution
When we solve a first order differential equation we will always start with a general solution. That is, a solution that includes some unknown constant C. The general solution will describe a family of curves.

6. Particular Solution
A particular solution is a solution that does not include an unknown constant C. To obtain a particular solution to a DE we need some additional information. We need an initial condition. An initial condition requires the curve of our solution to pass through a specific point (x0, y0). Initial conditions will be given in the form: y(x0) = y0 and will allow us to solve for the unknown constant C by plugging these values into our general solution and simplifying.

7. Separable Differential Equation
A separable differential equation is a class of DE (differential equation) that can be solved by separating the equation into a pair of integrals. 
We know that a DE is separable when it can be written as the product of two functions. First order separable differential equations will always have the following form:

8. Example
Is the following DE separable? Yes, because it can be interpreted as a function of x multiplied by the differential element dx, summed with a function of y multiplied by the differential element dy.

9. Example
Is the following DE separable? No, because it cannot be written in the following form:

10. How to Solve DEs Using Separation of Variables
To solve DEs using separation of variable we need to manipulate the equation into the form: After it is in this form, we integrate the left side with respect to y, and the right side with respect to x. The solution to these integrals will give us the general solution of the DE.

11. Generally:
In this example H(y) is the general solution to the DE.