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.
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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.
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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.
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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.
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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.
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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.
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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.
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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:
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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.
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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
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18. y‘=f(x,y) Linear Integrating Factor Non-Linear Separable Homogeneous
Change to Separable
Exact
Change to Exact
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19. Differential Equations
Solution of linear differential equations
We can then integrate both sides.
This will obtain the general solution.
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20. Differential Equations
Solution of linear differential equations 2.
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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.
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22. Using the product rule to differentiate the LHS we get:or
Using the product rule to differentiate the LHS we get:
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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
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24. 
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26. To solve this integration we need to use substitution.
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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) 
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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) 
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30. Hence the particular solution is
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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.
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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:
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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.
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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.
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35. Differential Equations
Solution of Separable differential equations
Separating the variables
For example:
so that:
That is:
Finally:
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36. Differential Equations
Solution of Separable differential equations
Separating the variables
For example:
so that:
Let y=zx; that is:
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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
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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
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39. Differential Equations
Solution of Inseparable differential equations
Cont. For example:
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40. Differential Equations
Solution of Inseparable differential equations
For example:
so that:
Let y=zx; that is: or
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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
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42. Differential Equations
Solution of Inseparable differential equations
Cont. For example:
Noting z = y/x, the solution is: or
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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.
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44. Differential Equations
Solution of Homogeneous differential equations
For example:
Solution:
Integrate
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45. Differential Equations
Solution of Homogeneous differential equations
For example:
Solution:
Let y=zx or
Separating the variables, we get
Integrating
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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)
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47. Differential Equations
Solution of Inhomogeneous differential equations 1)
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48. Differential Equations
Solution of Inhomogeneous differential equations 2)
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49. Differential Equations
Solution of Inhomogeneous differential equations 2)
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50. Differential Equations
Solution of Inhomogeneous differential equations 2)
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51. Differential Equations
Solution of Inhomogeneous differential equations
For example:
Solution:
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52. Differential Equations
Solution of Inhomogeneous differential equations
For example:
Solution:
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53. Differential Equations
Solution of Inhomogeneous differential equations
For example:
Solution:
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54. Differential Equations
Solution of Inhomogeneous differential equations
For example:
Solution:
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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
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57. Exactness Test
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58. STEP:1
STEP:2
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59. STEP:3
Find k(y)
STEP:4
STEP:5
STEP:6
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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.
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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
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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
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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
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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
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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
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67. Example 8 Test whether the following DE is exact. If exact, solve it.
Here
Hence exact.
Now
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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.
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69. IF
Making Exact
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70. Integrating Factor
To make DE Exact
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71. Integrating Factor
To make DE Exact
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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
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76. Multiplying by x2, the given DE becomes
which is of the form
Note that now
Integrating, we easily see that the solution is
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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
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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.
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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
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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.
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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:
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82. Differential Equations
Bernoulli differential equations
Substitution yields:
then:
becomes:
Which can be solved using the integrating factor method.
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83. Solution Example 1 Solve the following D.E.
Divide both sides by yn to give:
Let z = y1−n so that:
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84. Example 1 Solve the following D.E.
Solution
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