2. Aims: To be able to solve an explicit differential equation. (Basic – core 1 work!) To be able to find a general solution. To be able to find a particular solution. To be able to solve a differential equation by separating the variables. To recall integrating the exponential function. Differential Equations Lesson 1

3. Differential equations A differential equation in two variables x and y is an equation that contains derivatives of y with respect to x. For example: The simplest differential equations like are those of the form: D E’s of this form are solved by integrating both sides w. r. t x to give: For example, suppose we have the differential equation: 123 1

4. Differential equations of the form = f ( x ) Integrating both sides with respect to x gives: Since the constant c can take any value, this represents a whole family of solutions as shown here: This is called the G___________ Solution. If we are also told that when x = 1, y = 4. Then we can find what is called a P___________ Solution.

5. Separable variables like qu’s Differential equations that can be arranged in the form can be solved by the method of s_______________the variables. This method works by collecting all the terms in y, including the ‘ dy ’, on one side of the equation, and all the terms in x, including the ‘ dx ’, on the other side, and then integrating. 32 Find the general solution to. We only need a ‘ c ’ on one side of the equation. Rearrange to give:

6. Find the general solution to. Another example and see how we write the constant differently: Separable variables

8. Find the general solution to. On w/b

9. Find the general solution to. On w/b

10. Find the general solution to. On w/b

11. Find the general solution to. On w/b

12. Find the general solution to. On w/b

13. Find the general solution to. On w/b

14. Separable variables Separating the variables and integrating: Using the laws of indices this can be written as: Find the particular solution to the differential equation given that y = ln when x = 0.

15. Separable variables The particular solution is therefore: Given that y = ln when x = 0: Take the natural logarithms of both sides:

16. 1.Complete the matching puzzle 2. Do exercise C page 72 qu 1-5. Within the time limit Match all 30 cards – you’re as clever as Isaac Newton Match 25 or more – you’re as clever as Rachel Riley or Chen Jing-run. 中文 : 陈景润，中国数学家 Match 20 or more – you’re done a good job Match less than 9 – you need to spend more time looking at this topic and please come to a support session, as one of the maths team will be more than happy to help! START!