Differential Equations Separation of variables Exact factors of Differential Equations Integrating factors Before you begin you should be confident with: Integration by parts, substitution and trigonometric functions Differential Equations by separation of variables Implicit differentiation

Reminder Solve the following differential equations x=1, y=0 x=0, y=0 Integrate by parts

Try solving this differential equation When x=0 and y=1 We cannot separate the variables in the above, however, you may have noticed that the LHS is the exact derivative of x2y.. We can therefore write the equation as.. Substituting x=0, y=1

Solve the following differential equation… You may notice that… When x=1 and y=1

Now solve this differential equation Exact derivatives We were able to solve these two differential equations as the LHS were exact derivatives. Unfortunately this isn’t always the case… This is not an exact derivative of anything. What could I multiply it by to make it an exact derivative?? We call “x” the integrating factor Now solve this differential equation

Integrating Factors It is not always obvious what the integrating factor is but fortunately, we can work it out. Consider the differential equation.. Notice the coefficient of dy/dx is 1 We multiply by an integrating factor, f(x)

Notice the coefficient of dy/dx is 1 We multiply by an integrating factor, f(x)

Notice the coefficient of dy/dx is 1 We multiply by an integrating factor, f(x) The first term is one of the terms we get when we differentiate yf(x). We want the second term to be the other.

The Shizzle This is what we use to find the integrating factor

Example 1 Note: The coefficient of dy/dx must be 1 P(x) is the coefficient of y This is our integrating factor

Example 2 This is our integrating factor

Exercise 1. Find the general solution to the differential equation 2. Given that when x = 0, y = 1, solve the differential equation

Answers 1. Find the general solution to the differential equation 2. Given that when x = 0, y = 1, solve the differential equation

Links to other resources Mathsnet Q1 Mathsnet Q2 Mathsnet Q3 Mathsnet Q4 Mathsnet Q5