Introduction to Differential Equations and First-Order Differential Equations.pptx

Discussion on Differential Equation

1. STM 604 – Differential Equation Introduction to Differential Equations 1 STM 604 Differential Equation ELLVAN M. CAMPOS, MST Faculty, Institute of Advanced Studies Introduction to Differential Equations

2. Lesson 1: Definition and Terminologies 2 STM 604 Differential Equation Introduction to Differential Equations

3. Definition 3 STM 604 Differential Equation Definition Classification Solution of an ODE Examples Consider this: This is the same with or Solution: function(s)   Introduction to Differential Equations The difference of this kind of equation: or Solution: values(s)   An equation containing the derivatives of one or more dependent variables with respect to one or more independent variables is a differential equation (DE).

4. Classification (by Type) If an equation contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable, it is said to be an ordinary differential equation (ODE) .   4 STM 604 Differential Equation Introduction to Differential Equations Definition Classification Solution of an ODE Examples

5. Classification (by Type) 5 STM 604 Differential Equation If an equation involving the partial derivatives of one or more dependent variables of two or more independent variables, it is said to be an partial differential equation (PDE) .   Introduction to Differential Equations Definition Classification Solution of an ODE Examples

6. Classification (by Order) The order of a DE is the order of the highest derivative in the equation.   6 STM 604 Differential Equation 2 nd order 1 st order Introduction to Differential Equations Definition Classification Solution of an ODE Examples

7. Classification (by Linearity) 7 STM 604 Differential Equation An n th-order ordinary differential equation is said to be linear if F is linear in , in form: or Two Properties of Linear Differential Equation: (1) The dependent variable and all its derivatives are of the first degree; and (2) Each coefficient depends at most on the independent variable x.   Introduction to Differential Equations Definition Classification Solution of an ODE Examples

8. Classification (by Linearity) 8 STM 604 Differential Equation Samples   Non-Samples   Introduction to Differential Equations Definition Classification Solution of an ODE Examples

9. Solution of an ODE 9 STM 604 Differential Equation Any function , defined on an interval I and possessing at least n derivatives that are continuous on I , which when substituted into an n th-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval. From this definition, the interval I is variously called as interval of definition of the solution and can be an open interval , a closed interval , an infinite interval , and so on.   Introduction to Differential Equations Definition Classification Solution of an ODE Examples

10. Example: Verification of Solution 10 STM 604 Differential Equation Verify that the indicated function is a solution of the given differential equation on the interval . Note: is a constant solution under the interval given. A solution of a differential equation that is identically zero on an interval I is said to be a trivial solution .   Introduction to Differential Equations Definition Classification Solution of an ODE Examples

11. Example: Verification of Solution 11 STM 604 Differential Equation Verify that the indicated function is a solution of the given differential equation on the interval . Note: is a constant solution under the interval given. A solution of a differential equation that is identically zero on an interval I is said to be a trivial solution .   Introduction to Differential Equations Definition Classification Solution of an ODE Examples

12. Lesson 2: Separable Variables 12 STM 604 Differential Equation First-Order Differential Equations

13. Definition 13 STM 604 Differential Equation Definition Samples A first-order differential equation of the form is said to be separable or to have separable variables . The equations and are separable and nonseparable respectively. Method of Solution : Given that , integrate the following separately.   First-Order Differential Equations

14. Samples 1. Determine if the differential equation is separable. If it is, solve the DE using separation of variables.   14 STM 604 Differential Equation First-Order Differential Equations Definition Samples

15. Samples 2. Determine if the differential equation is separable. If it is, solve the DE using separation of variables. Answer:   15 STM 604 Differential Equation First-Order Differential Equations Definition Samples

16. Samples 3. Determine if the differential equation is separable. If it is, solve the DE using separation of variables, and find the specific equation if . Answer:   16 STM 604 Differential Equation First-Order Differential Equations Definition Samples

17. Samples 4. Determine if the differential equation is separable. If it is, solve the DE using separation of variables. Answer:   17 STM 604 Differential Equation First-Order Differential Equations Definition Samples

18. Lesson 3: Linear Equations 18 STM 604 Differential Equation First-Order Differential Equations

19. Definition 19 STM 604 Differential Equation A first-order differential equation of the form is said to be linear equation . When , the linear equation is said to be homogenous; otherwise, it is nonhomogenous .   First-Order Differential Equations Definition Methods of Solution Samples

20. Solving a Linear First-Order Equation 20 STM 604 Differential Equation From , divide to obtain the standard form of the linear equation, given by . From the standard form, identify and then find the integrating factor Multiply the standard form of the equation by the integrating factor. The left-hand side of the resulting equation is automatically the derivative of the integrating factor and y: Integrate both sides of this last equation.   First-Order Differential Equations Definition Methods of Solution Samples

21. Samples 1. Solve the differential equation .   21 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples

22. Samples 2. Solve the differential equation . Answer:   22 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples

23. Samples 3. Solve the differential equation . Answer:   23 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples

24. Lesson 4: Exact Equations 24 STM 604 Differential Equation First-Order Differential Equations

25. Differential of a Function of Two Variables 25 STM 604 Differential Equation If is a function of two variables with continuous partial derivatives, it’s (total) differential is . If , then . For example, given , then . So, is it equivalent to the differential ?   First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

26. Definition 26 STM 604 Differential Equation A differential expression is an exact differential if it corresponds to the differential of some function . A first-order differential equation of the form is said to exact equation if the expression on the left-hand side is an exact differential.   First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

27. Theorem 27 STM 604 Differential Equation Let and be continuous and have continuous first partial derivatives, then a necessary sufficient condition that be exact differential is .   First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

28. Method of Solution 28 STM 604 Differential Equation Given the differential equation , determine whether holds. If exists, express function for which Find by integrating with respect to , while holding constant: , where the arbitrary function is the “constant” of integration. Differentiate with respect to and assume . This gives . Integrate with respect to and substitute the result to .   First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

29. Samples 1. Solve the differential equation .   29 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

30. Samples 2. Solve the differential equation Answer:   30 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

31. Non-Exact Differential Equations 31 STM 604 Differential Equation For non-exact DE, look for an integrating factor to multiply to the given DE. Given if is a function of alone, then an integrating factor for the given DE is if is a function of alone, then an integrating factor for the given DE is   First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

32. Samples 3. Verify is the given DE is not exact. Multiply the given DE by the indicated integrating factor and verify that the new equation is exact. Solve the new DE.   32 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

33. Lesson 5: Solutions by Substitution 33 STM 604 Differential Equation First-Order Differential Equations

34. Homogenous First-Order Differential Equations A first-order DE in the form is said to be homogenous if both and are homogenous functions of the same degree. Thus, and . Substitutions may be on the following ways: Either of the substitutions or , where and are new dependent variables. For differential equation of the form can be reduced to an equation by means of . These two ways reduces the differential equations to separation of variables.   34 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

35. Samples 1. Solve the differential equation .   35 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

36. Samples 2. Solve the differential equation . Answer:   32 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

37. Samples 3. Find the explicit solution of the differential equation ;   32 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

38. Samples 4. Solve the differential equation Answer:   32 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE

39. - End  Thank you! 33 STM 604 Differential Equation