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Engineering Analysis I

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1 Engineering Analysis I
Basics Concepts and Ideas First Order Differential Equations Dr. Omar R. Daoud

2 Differential Equations
Many Physical laws and relations appear mathematically in the form of Differentia Equations  They are one of the important fundamentals in engineering mathematics. An ordinary differential equation is an equation that contains one or several derivatives of unknown functions A differential equation is a relationship between an independent variable x, a dependent variable y and one or more derivatives of y with respect to x. If the unknown function depends only on one independent variable, then its called by Ordinary D.E (O.D.E.), while its denoted by Partial D.E. (P.D.E) if the function depends on two or more independent variables. 11/28/2018 Part I

3 Differential Equations
The order of a differential equation is given by the highest derivative involved. A function which satisfies the equation is called a solution to the differential equation. Solving a differential equation is the reverse process to the one just considered. To solve a differential equation a function has to be found for which the equation holds true. The solution will contain a number of arbitrary constants – the number equalling the order of the differential equation. 11/28/2018 Part I

4 Differential Equations
Linear vs. nonlinear differential equations A linear differential equation contains only terms that are linear in both the dependent variable and its derivatives. A nonlinear differential equation contains nonlinear function of the dependent variable. 11/28/2018 Part I

5 ordinary differential equations
Definition: A differential equation is an equation containing an unknown function and its derivatives. Examples:. 1. 2. 3. y is dependent variable and x is independent variable, and these are ordinary differential equations

6 Partial Differential Equation
Examples: 1. u is dependent variable and x and y are independent variables, and is partial differential equation. 2. 3. u is dependent variable and x and t are independent variables

7 Order of Differential Equation
The order of the differential equation is order of the highest derivative in the differential equation. Differential Equation ORDER 1 2 3

8 Degree of Differential Equation
The degree of a differential equation is power of the highest order derivative term in the differential equation. Differential Equation Degree 1 1 3

9 Linear Differential Equation
A differential equation is linear, if 1. dependent variable and its derivatives are of degree one, 2. coefficients of a term does not depend upon dependent variable. Example: 1. is linear. Example: 2. is non - linear because in 2nd term is not of degree one.

10 3. 4. is non – linear Example: Example:
is non - linear because in 2nd term coefficient depends on y. 3. Example: 4. is non – linear is non - linear because

11 Differential Equations
Homogeneous vs. Inhomogeneous differential equations A linear differential equation is homogeneous if every term contains the dependent variable or its derivatives. A homogeneous differential equation can be written as where L is a linear differential operator. A homogeneous differential equation always has a trivial solution y(x) = 0. 11/28/2018 Part I

12 Differential Equations
An inhomogeneous differential equation has at least one term that contains no dependent variable. The general solution to a linear inhomogeneous differential equation can be written as the sum of two parts: Here yh(x) is the general solution of the corresponding homogeneous equation, and yp(x) is any particular solution of the inhomogeneous equation. 11/28/2018 Part I

13 Differential Equations
Thus, the first order differential equitation contains y’ and may contains y and given function of x; A solution of a given F.O.D.E. on some open interval a<x<b is a function y=h(x) that has a derivative which satisfies Equation (1) for all x in that interval. We can then integrate both sides. This will obtain the general solution. 11/28/2018 Part I

14 Differential Equations
Modelling is the steps that lead from the physical situation to a mathematical formulation and solution and to the Physical interpretation of the result; setting up a mathematical model (Differential Equations) of the physical process. Solving the D.Es. Determination of a particular solution from an initial conditions (to transform the general solution to a particular one).  An initial value problem is a differential equation together with an initial condition. Checking. 11/28/2018 Part I

15 Differential Equations
Formation of differential equations Differential equations may be formed from a consideration of the physical problems to which they refer. Mathematically, they can occur when arbitrary constants are eliminated from a given function. For example, let: 11/28/2018 Part I

16 Differential Equations
Formation of differential equations Here the given function had two arbitrary constants:  and the end result was a second order differential equation: In general an nth order differential equation will result from consideration of a function with n arbitrary constants. 11/28/2018 Part I

17 Differential Equations
Various methods of solving F.O.D.E. will be discussed. These include: Variables separable Homogeneous equations Exact equations Equations that can be made exact by multiplying by an integrating factor 11/28/2018 Part I

18 y‘=f(x,y) Linear Integrating Factor Non-Linear Separable Homogeneous
Change to Separable Exact Change to Exact 11/28/2018 Part I

19 Differential Equations
Solution of linear differential equations We can then integrate both sides. This will obtain the general solution. 11/28/2018 Part I

20 Differential Equations
Solution of linear differential equations 2. 11/28/2018 Part I

21 Differential Equations
Solution of linear differential equations 3. A first order linear differential equation is an equation of the form 1 To find a method for solving this equation, lets consider the simpler equation Which can be solved by separating the variables. 11/28/2018 Part I

22 Using the product rule to differentiate the LHS we get:
or Using the product rule to differentiate the LHS we get: 11/28/2018 Part I

23 If we multiply both sides by
Returning to equation 1, If we multiply both sides by Now integrate both sides. For this to work we need to be able to find 11/28/2018 Part I

24 11/28/2018 Part I

25 11/28/2018 Part I

26 To solve this integration we need to use substitution.
11/28/2018 Part I

27 (note the shortcut I have taken here)
The modulus vanishes as we will have either both positive on either side or both negative. Their effect is cancelled. (note the shortcut I have taken here) 11/28/2018 Part I

28 (note the shortcut I have taken here)
The modulus vanishes as we will have either both positive on either side or both negative. Their effect is cancelled. (note the shortcut I have taken here) 11/28/2018 Part I

29 11/28/2018 Part I

30 Hence the particular solution is
11/28/2018 Part I

31 Differential Equations
Solution of Separable differential equations F.O.D.E. can be reduced to the form of This equation is called separable because that the variables x and y could be separated, so that x appears only on one side while y on the other one. 11/28/2018 Part I

32 Differential Equations
Solution of Separable differential equations Direct integration If the differential equation to be solved can be arranged in the form: the solution can be found by direct integration. That is: 11/28/2018 Part I

33 Differential Equations
Solution of Separable differential equations Direct integration For example: so that: This is the general solution (or primitive) of the differential equation. If a value of y is given for a specific value of x then a value for C can be found. This would then be a particular solution of the differential equation. 11/28/2018 Part I

34 Differential Equations
Solution of Separable differential equations Separating the variables If a differential equation is of the form: Then, after some manipulation, the solution can be found by direct integration. 11/28/2018 Part I

35 Differential Equations
Solution of Separable differential equations Separating the variables For example: so that: That is: Finally: 11/28/2018 Part I

36 Differential Equations
Solution of Separable differential equations Separating the variables For example: so that: Let y=zx; that is: 11/28/2018 Part I

37 Differential Equations
Solution of Separable differential equations Separating the variables For example: Separating the variables, we get Integrating we get the solution as or 11/28/2018 Part I

38 Differential Equations
Solution of Inseparable differential equations Certain Des are inseparable, however they could be transferred to separable DE by the introduction of a new unknown function, For example: Divide by 2xy, then 11/28/2018 Part I

39 Differential Equations
Solution of Inseparable differential equations Cont. For example: 11/28/2018 Part I

40 Differential Equations
Solution of Inseparable differential equations For example: so that: Let y=zx; that is: or 11/28/2018 Part I

41 Differential Equations
Solution of Inseparable differential equations Cont. For example: Separating the variables, we get , Integrating we get We express the LHS integral by partial fractions. We get or 11/28/2018 Part I

42 Differential Equations
Solution of Inseparable differential equations Cont. For example: Noting z = y/x, the solution is: or 11/28/2018 Part I

43 Differential Equations
Homogenous differential equations The general form of the L.D.E. is If then the L.D.E. is called Homogenous otherwise it is not. The solution of Homogenous D.E. could be attained by using the separable method. The Integrating Factor (IF) will be used to change the Inhomogeneous D.E. to a homogenous D.E. 11/28/2018 Part I

44 Differential Equations
Solution of Homogeneous differential equations For example: Solution: Integrate 11/28/2018 Part I

45 Differential Equations
Solution of Homogeneous differential equations For example: Solution: Let y=zx or Separating the variables, we get Integrating 11/28/2018 Part I

46 Two Different ways to solve it
Differential Equations Solution of Inhomogeneous differential equations Two Different ways to solve it a0(x)=a’1(x) a0(x)≠a’1(x) 11/28/2018 Part I

47 Differential Equations
Solution of Inhomogeneous differential equations 1) 11/28/2018 Part I

48 Differential Equations
Solution of Inhomogeneous differential equations 2) 11/28/2018 Part I

49 Differential Equations
Solution of Inhomogeneous differential equations 2) 11/28/2018 Part I

50 Differential Equations
Solution of Inhomogeneous differential equations 2) 11/28/2018 Part I

51 Differential Equations
Solution of Inhomogeneous differential equations For example: Solution: 11/28/2018 Part I

52 Differential Equations
Solution of Inhomogeneous differential equations For example: Solution: 11/28/2018 Part I

53 Differential Equations
Solution of Inhomogeneous differential equations For example: Solution: 11/28/2018 Part I

54 Differential Equations
Solution of Inhomogeneous differential equations For example: Solution: 11/28/2018 Part I

55 Differential Equations
Exact differential equations A F.O.D.E is called an Exact if there exits a function f(x, y) such that Here df is the ‘total differential’ of f(x, y) and equals Hence the given DE becomes df = 0 Integrating, we get the solution as f(x, y) = C 11/28/2018 Part I

56 11/28/2018 Part I

57 Exactness Test 11/28/2018 Part I

58 STEP:1 STEP:2 11/28/2018 Part I

59 STEP:3 Find k(y) STEP:4 STEP:5 STEP:6 11/28/2018 Part I

60 11/28/2018 Part I

61 The integral of zero is ZERO, simple
The integral of zero is ZERO, simple. Although derivative of a constant would be zero, but integral of zero would always be zero. One thing to note: Integral is NOT antiderivative in strict sense. Its an area under graph f(x) in Cartesian system, where it is ofcourse in a two dimensional plane. 11/28/2018 Part I

62 Example 6 Test whether the following DE is exact. If exact, solve it.
Here Hence exact. Now Differentiating partially w.r.t. y, we get Hence 11/28/2018 Part I

63 Example 6 Test whether the following DE is exact. If exact, solve it.
Integrating, we get (Note that we have NOT put the arb constant ) Hence Thus the solution of the given D.E. is or 11/28/2018 Part I

64 Example 7 Test whether the following DE is exact. If exact, solve it.
Here Hence exact. Now Differentiating partially w.r.t. y, we get Hence 11/28/2018 Part I

65 Thus the solution of the given d.e. is
Integrating, we get Hence Thus the solution of the given d.e. is c an arb const. or 11/28/2018 Part I

66 In the above problems, we found f(x, y) by integrating M partially w.r.t. x and then equated
We can reverse the roles of x and y. That is we can find f(x, y) by integrating N partially w.r.t. y and then equate 11/28/2018 Part I

67 Example 8 Test whether the following DE is exact. If exact, solve it.
Here Hence exact. Now 11/28/2018 Part I

68 Differentiating partially w.r.t. x, we get
gives Integrating, we get Hence Thus the solution of the given D.E. is or c an arb const. 11/28/2018 Part I

69 IF Making Exact 11/28/2018 Part I

70 Integrating Factor To make DE Exact 11/28/2018 Part I

71 Integrating Factor To make DE Exact 11/28/2018 Part I

72 11/28/2018 Part I

73 11/28/2018 Part I

74 11/28/2018 Part I

75 Example 3 Find an I.F. for the following DE and hence solve it.
Here Hence the given DE is not exact. Now a function of x alone. Hence is an integrating factor of the given DE 11/28/2018 Part I

76 Multiplying by x2, the given DE becomes
which is of the form Note that now Integrating, we easily see that the solution is 11/28/2018 Part I

77 Example 4 Find an I.F. for the following DE and hence solve it.
Here Hence the given DE is not exact. Now a function of y alone. Hence 11/28/2018 Part I

78 is an integrating factor of the given DE
Multiplying by sin y, the given DE becomes which is of the form Note that now Integrating, we easily see that the solution is c an arbitrary constant. 11/28/2018 Part I

79 Example 5 Find an I.F. for the following DE and hence solve it.
Here Hence the given DE is not exact. Now a function of z =x y alone. Hence 11/28/2018 Part I

80 the given DE becomes is an integrating factor of the given DE
Multiplying by the given DE becomes which is of the form Integrating, we easily see that the solution is c an arbitrary constant. 11/28/2018 Part I

81 Differential Equations
Bernoulli differential equations A Bernoulli equation is a differential equation of the form: This is solved by: Divide both sides by yn to give: Let z = y1−n so that: 11/28/2018 Part I

82 Differential Equations
Bernoulli differential equations Substitution yields: then: becomes: Which can be solved using the integrating factor method. 11/28/2018 Part I

83 Solution Example 1 Solve the following D.E.
Divide both sides by yn to give: Let z = y1−n so that: 11/28/2018 Part I

84 Example 1 Solve the following D.E.
Solution 11/28/2018 Part I


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