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Lecture 1

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Lecture 1

1. 1. CHAPTER 2 Differential Equation
2. 2. Introduction Consider x as an independent variable and y as dependent variable. An equation that involves at least one derivative of y with respect to x, e.g. Is known as a differential equation or common differential equation .
3. 3. b) c) d) a) Example of Differential Equation
4. 4. <ul><li>Order is the highest derivative </li></ul><ul><li>Degree is the highest power of the highest </li></ul><ul><li>derivative </li></ul><ul><li>Examples: a) </li></ul>This DE has order 2 (the highest derivative appearing is the second derivative) and degree 1 (the power of the highest derivative is 1.) Order & Degree
5. 5. <ul><li>In this chapter we only deal with first order, </li></ul><ul><li>first degree differential equations. </li></ul><ul><li>A solution for a differential equation is a </li></ul><ul><li>function whose elements and derivatives may be </li></ul><ul><li>substituted into the differential equation. There </li></ul><ul><li>are two types of solution for differential </li></ul><ul><li>equations </li></ul>
6. 6. <ul><li>General solution – The general solution of a differential equation contains an arbitrary constant c . </li></ul><ul><li>Particular solution - The particular solution of a differential equation contains a specified initial value and containing no constant. </li></ul>Solutions
7. 7. Examples Of General Solution <ul><li>This is already in the required form, so we </li></ul><ul><li>simply integrate: </li></ul><ul><li>, c is constant </li></ul>
8. 8. Example First we must separate the variables: Multiply throughout by dx Divide throughout by y Divide throughout by x
9. 9. This gives us: We now integrate ,
10. 10. Example First we must separate the variables: Multiply throughout by dx Divide throughout by Divide throughout by
11. 11. <ul><li>This gives us: </li></ul>We now integrate:
12. 12. Separable Variables and Integrating Factors <ul><li>Example </li></ul><ul><li>Solve the differential equation </li></ul>First we must separate the variables: Consider :
13. 13. <ul><li>By using substitution, </li></ul>
14. 14. <ul><li>Consider </li></ul><ul><li>Let </li></ul>and By using integration by part
15. 15. The General Solution :
16. 16. Example <ul><li>Solve the differential equation </li></ul><ul><li>when x =0, y=5 </li></ul><ul><li>First we must separate the variables: </li></ul>We now integrate:
17. 17. <ul><li>We now use the information which means </li></ul><ul><li>at , to find c . </li></ul>So the particular solution is: gives
18. 18. Exercise : Solve the initial value problem. Express the solution implicitly. a. b.