1. Engineering Analysis I
Second and Higher Order Differential Equations
Dr. Omar R. Daoud

2. 2nd Order Differential Equations
Introduction
2nd Order Differential Equations
For any three numbers a, b and c, the two numbers:
are solutions to the quadratic equation:
with the properties:
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3. 2nd Order Differential Equations
Introduction
2nd Order Differential Equations
The differential equation:
can be re-written to read:
that is:
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4. 2nd Order Differential Equations
Introduction
2nd Order Differential Equations
The differential equation can again be re-written as:
where:
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5. 2nd Order Differential Equations
Introduction
2nd Order Differential Equations
The differential equation:
has solution:
This means that:
That is:
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6. 2nd Order Differential Equations
Introduction
2nd Order Differential Equations
The differential equation:
has solution:
where:
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7. 2nd Order Differential Equations
Introduction
2nd Order Differential Equations
A 2nd Order D.E. will appear in the form of
It will be denoted by homogeneous if otherwise its not.
Moreover, if one of the solutions is known
 The reduction of order technique will be used to find the second linearly independent solution
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8. 2nd Order Differential Equations
The Reduction of order technique
2nd Order Differential Equations
Assume that the 2nd order D.E. has two different basis (fundamental solutions); namely y1 and y2.
They called linearly independent on a certain set I if and only if these basis are not proportional;
while they called linearly dependent if
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9. 2nd Order Differential Equations
The Reduction of order technique
2nd Order Differential Equations
To obtain the basis if y1(x) is known, y2(x) should be set to
As its known that the homogenous 2nd order D.E. is in the form of
However,
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10. 2nd Order Differential Equations
The Reduction of order technique
2nd Order Differential Equations
Substitute 1 and 3 in 2
Rearranging and simplifying
Let then 5 will be
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11. 2nd Order Differential Equations
The Reduction of order technique
Integrate
Take the exponential and manipulate
2nd Order Differential Equations
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12. 2nd Order Differential Equations
The Reduction of order technique
Example: Find the basis for the following D.E.
Divide both sides by x2
Then
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13. 2nd Order Differential Equations
The Reduction of order technique
Example: Find the basis for the following D.E.
Divide both sides by x2
Then
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14. 2nd Order Differential Equations
Homogeneous equations
2nd Order Differential Equations
The differential equation:
Is a second-order, constant coefficient, linear, homogeneous differential equation. Its solution is found from the solutions to the auxiliary equation:
These are:
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15. 2nd Order Differential Equations
The auxiliary equation
Real and different roots
Real and equal roots
Complex roots
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16. 2nd Order Differential Equations
The auxiliary equation
Real and different roots
If the auxiliary equation:
with solution:
where:
then the solution to:
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17. 2nd Order Differential Equations
The auxiliary equation
Real and different roots
Example: Find the general solution for the following D.E.
The auxiliary equation is
Thus the general solution is
P.S. To find a particular solution we must be given enough information.
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18. 2nd Order Differential Equations
The auxiliary equation
Real and different roots
Example: Find the general solution for the following D.E.
The auxiliary equation is
Thus the general solution is
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19. 2nd Order Differential Equations
The auxiliary equation
Real and different roots
Cont. Example: Find the general solution for the following D.E.
Using the initial conditions
Now
Solving gives
Thus the particular solution is
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20. 2nd Order Differential Equations
The auxiliary equation
Real and equal roots
If the auxiliary equation:
with solution:
where:
then the solution to:
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21. 2nd Order Differential Equations
The auxiliary equation
Real and equal roots
Example: Find the general solution for the following D.E.
The auxiliary equation is
Thus the general solution is
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22. 2nd Order Differential Equations
The auxiliary equation
Real and equal roots
Example: Find the general solution for the following D.E.
The auxiliary equation is 
Thus the general solution is
Using the initial conditions
Now
Thus the particular solution is
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23. 2nd Order Differential Equations
The auxiliary equation
Complex roots
If the auxiliary equation:
with solution:
where:
Then the solutions to the auxiliary equation are complex conjugates. That is:
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24. 2nd Order Differential Equations
The auxiliary equation
Complex roots
Complex roots to the auxiliary equation:
means that the solution of the differential equation:
is of the form:
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25. 2nd Order Differential Equations
The auxiliary equation
Complex roots
Since:
then:
The solution to the differential equation whose auxiliary equation has complex roots can be written as::
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26. 2nd Order Differential Equations
The auxiliary equation
Complex roots
Example: Find the general solution for the following D.E.
The auxiliary equation is
Thus the general solution is
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27. 2nd Order Differential Equations
The auxiliary equation
Complex roots
Example: Find the general solution for the following D.E.
The auxiliary equation is
Thus the general solution is
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28. 2nd Order Differential Equations
The auxiliary equation
Complex roots
Cont. Example: Find the general solution for the following D.E.
Using the initial conditions 
Thus the particular solution is
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29. 2nd Order Differential Equations
Summary
Differential equations of the form:
Auxiliary equation:
Roots real and different: Solution
Roots real and the same: Solution
Roots complex (  j): Solution
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30. 2nd Order Differential Equations
Differential Operators (Method of inverse operators)
Sometimes, it is convenient to refer to the symbol “D” as the differential operator:
But,
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31. 2nd Order Differential Equations
Differential Operators (Method of inverse operators)
The differential operator D can be treated as an ordinary algebraic quantity with certain limitations.
(1) The distribution law:
A(B+C) = AB + AC
which applies to the differential operator D
(2) The commutative law:
AB = BA
which does not in general apply to the differential operator D
Dxy  xDy
(D+1)(D+2)y = (D+2)(D+1)y
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32. 2nd Order Differential Equations
Differential Operators (Method of inverse operators)
The differential operator D can be treated as an ordinary algebraic quantity with certain limitations.
(3) The associative law:
(AB)C = A(BC)
which does not in general apply to the differential operator D
D(Dy) = (DD)y
D(xy) = (Dx)y + x(Dy)
The basic laws of algebra thus apply to the pure operators, but the
relative order of operators and variables must be maintained.
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33. 2nd Order Differential Equations
Differential Operators to exponentials
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34. 2nd Order Differential Equations
Differential Operators to trigonometric functions
where “Im” represents the imaginary part of the function which follows it.
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35. 2nd Order Differential Equations
Differential Operators
The operator D signifies differentiation, i.e.
D-1 is the “inverse operator” and is an “intergrating” operator.
It can be treated as an algebraic quantity in exactly the same manner as D
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36. 2nd Order Differential Equations
Differential Operators
Example: Find the general solution for the following D.E. Using the Differential Operators method
The Differential Operators
Then the general solution is
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37. 2nd Order Differential Equations
Euler-Cauchy Equation
It is a differential Equation in the following form
The basic method of solution
Characteristic equation
1.two distinct, real values for r (when (A-1)2-4B > 0)
2.Only one real value for r (when (A-1)2-4B = 0)
3.Two distinct, complex values for r (when (A-1)2-4B < 0)
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38. Euler - Cauchy equation
Case 1 : Two distinct, real values for r (when (A-1)2-4B > 0)
The general solution :
Where c1 and c2 are arbitrary constants
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39. 2nd Order Differential Equations
Euler-Cauchy Equation
Example: Solve the following D.E.
Solution
Then the general solution is
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40. Euler - Cauchy equation
Case 2 : Only one real value for r (when (A-1)2-4B = 0)
To find y2(x), we apply a technique called reduction of order
Try to produce u(x) such that u(x) y1(x) is a solution.
Substituted into
The general solution :
Where c1 and c2 are arbitrary constants
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41. 2nd Order Differential Equations
Euler-Cauchy Equation
Example: Solve the following D.E.
Solution
Then the general solution is
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42. Where c1 and c2 are arbitrary constants
Euler - Cauchy equation
Case 3 : Two distinct, complex values for r (when (A-1)2-4B < 0)
Any linear combination of solutions is also a solution !
The general solution :
Where c1 and c2 are arbitrary constants
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43. 2nd Order Differential Equations
Euler-Cauchy Equation
Example: Solve the following D.E.
Solution
Then the general solution is
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44. 2nd Order Differential Equations
Inhomogeneous equations
The second-order, constant coefficient, linear, inhomogeneous differential
equation is an equation of the type:
The solution is in two parts y1 + y2:
part 1, y1 is the solution to the homogeneous equation and is called the complementary function which is the solution to the homogeneous equation
part 2, y2 is called the particular integral.
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45. 2nd Order Differential Equations
Inhomogeneous equations
Particular integrals
The general form assumed for the particular integral depends upon the form of
the right-hand side of the inhomogeneous equation. The following table can be
used as a guide:
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46. 2nd Order Differential Equations
Inhomogeneous equations
Example: Find the general solution for the following D.E.
The auxiliary equation is
Thus the CF is
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47. 2nd Order Differential Equations
Inhomogeneous equations
Cont. Example: Find the general solution for the following D.E.
Finding the PI:
Substituting into the original equation
Thus the PI is
Hence the general solution is
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48. 2nd Order Differential Equations
Inhomogeneous equations
Complementary function
Example, to solve:
Complementary function
Auxiliary equation: m2 – 5m + 6 = 0 solution m = 2, 3
Complementary function y1 = Ae2x + Be3x where:
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49. 2nd Order Differential Equations
Inhomogeneous equations
Particular integral
(b) Particular integral
Assume a form for y2 as y2 = Cx2 + Dx + E then substitution in:
gives:
yielding:
so that:
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50. complementary function + particular integral
2nd Order Differential Equations
Inhomogeneous equations
Complete solution
(c) The complete solution to:
consists of:
complementary function + particular integral
That is:
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51. Thus we should try another y(x)
2nd Order Differential Equations
Inhomogeneous equations
Example: Find the general solution for the following D.E.
The auxiliary equation is
Thus the CF is
Finding the PI:
But this is in the same form of one of the CF’s solutions.
Thus we should try another y(x)
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52. 2nd Order Differential Equations
Inhomogeneous equations
Cont. Example: Find the general solution for the following D.E.
Thus, PI should be:
Substituting into the original equation
Equating Coefficients 
Hence the general solution is
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53. 2nd Order Differential Equations
Inhomogeneous equations
Solution by undetermined Coefficients
Example: Solve the following D.E.:
Solution:
The auxiliary equation is
Thus the CF is
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54. 2nd Order Differential Equations
Inhomogeneous equations
Solution by undetermined Coefficients
Example: Solve the following D.E.:
Solution:
Finding the PI: (Back to slide 45)
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55. 2nd Order Differential Equations
Inhomogeneous equations
Solution by undetermined Coefficients
Cont. Example: Solve the following D.E.:
Solution:
Finding the PI: (Back to slide 45)
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56. 2nd Order Differential Equations
Inhomogeneous equations
Solution by undetermined Coefficients
Cont. Example: Solve the following D.E.:
Solution:
Finding the PI: (Back to slide 45)
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57. 2nd Order Differential Equations
Inhomogeneous equations
Solution by undetermined Coefficients
Cont. Example: Solve the following D.E.:
Solution:
Finding the PI: (Back to slide 45)
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58. 2nd Order Differential Equations
Inhomogeneous equations
Solution by undetermined Coefficients
Cont. Example: Solve the following D.E.:
Solution:
Apply the conditions:
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59. 2nd Order Differential Equations
Inhomogeneous equations
Example: Find the general solution for the following D.E. Using the Differential Operators method
The Differential Operators
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60. 2nd Order Differential Equations
Inhomogeneous equations
Cont. Example: Find the general solution for the following D.E. Using the Differential Operators method
Or, The Differential Operators
binomial expansion
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61. 2nd Order Differential Equations
Inhomogeneous equations
Cont. Example: Find the general solution for the following D.E. Using the Differential Operators method =2
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62. 2nd Order Differential Equations
Conclusion
If f(p) = 0, Use dimensional analysis
Non-zero
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63. 2nd Order Differential Equations
Conclusion
y = 1, p = 0, Upcoming(D-p) replaced byD
integration
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64. 2nd Order Differential Equations
Inhomogeneous equations: Solution by undetermined Coefficients or or or
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65. 2nd Order Differential Equations
Inhomogeneous equations
Solution by Variation of Parameters
The Linear inhomogeneous 2nd order D.E. will be in the following form:
here: p(x) and q(x) need not to be constant
Let y1(x) and y2(x) are solutions of y’’+ p(x)y’ + q(x)y = 0 on the interval J.
Let yp(x) = u(x)y1(x) + v(x)y2(x) 
But
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66. 2nd Order Differential Equations
Inhomogeneous equations
Solution by Variation of Parameters
Thus 
Then
Substitute them in the general equation
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67. 2nd Order Differential Equations
Inhomogeneous equations
Solution by Variation of Parameters
Thus 
yp(x) = u(x)y1(x) + v(x)y2(x)
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68. 2nd Order Differential Equations
Inhomogeneous equations (Solution by Variation of Parameters)
Example: Find the general solution for the following D.E.
Solution:
The auxiliary equation is
Thus the CF is
The Wronskian factor is
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69. 2nd Order Differential Equations
Inhomogeneous equations (Solution by Variation of Parameters)
Cont. Example: Find the general solution for the following D.E.
Solution:
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70. 2nd Order Differential Equations
Applications:
Ex1:
Consider the following interconnected fluid tanks:
Suppose both tanks, each holding 24 liters of a brine solution, are interconnected by pipes as shown. Fresh water flows into tank A at a rate of 6 L/min, and fluid is drained out of tank B at the same rate; also 8 L/min of fluid are pumped from tank A to tank B, and 2 L/min from tank B to tank A.
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71. 2nd Order Differential Equations
Applications:
Ex1:
Consider the following interconnected fluid tanks:
The liquids inside each tank are kept well stirred, so that each mixture is homogeneous. If initially tank A contains a kg of salt and tank B contains b kg of salt, determine the mass of salt in each tanks at any time t > 0.
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72. 2nd Order Differential Equations
Applications:
Ex1:
Consider the following interconnected fluid tanks:
For tank A, we have:
For tank B, we have:
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73. This gives us a system of First Order Equations
2nd Order Differential Equations
Applications:
Ex1:
This gives us a system of First Order Equations
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74. 2nd Order Differential Equations
Applications:
Ex2:
Two cases of RLC Circuit
The total charges at any time in these two circuits
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75. 2nd Order Differential Equations
Applications:
Ex2:
Two cases of RLC Circuit
The total charges at any time in these two circuits
The voltage drop across R
The voltage drop across L
The voltage drop across C
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76. 2nd Order Differential Equations
Applications:
Ex2:
Two cases of RLC Circuit
The total charges at any time in these two circuits
The total voltage drop is
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77. 2nd Order Differential Equations
Applications:
Ex2:
Two cases of RLC Circuit
The total charges at any time in these two circuits
The total voltage drop is
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78. 2nd Order Differential Equations
Applications:
Ex2:
Two cases of RLC Circuit
Find the current I(t) in the given RLC circuit with R=100Ω, L=0.1 H, C=1mF, and E(t)=155sin(377t)V. Q(0)=I(0)=0.
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79. 2nd Order Differential Equations
Applications:
Ex2:
Two cases of RLC Circuit
Find the current I(t) in the given RLC circuit with R=100Ω, L=0.1 H, C=1mF, and E(t)=155sin(377t)V. Q(0)=I(0)=0.
The auxiliary equation is
The CF solution
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80. 2nd Order Differential Equations
Applications:
Ex2:
Two cases of RLC Circuit
Find the current I(t) in the given RLC circuit with R=100Ω, L=0.1 H, C=1mF, and E(t)=155sin(377t)V. Q(0)=I(0)=0.
The Ip is
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81. 2nd Order Differential Equations
Applications:
Ex2:
Two cases of RLC Circuit
Find the current I(t) in the given RLC circuit with R=100Ω, L=0.1 H, C=1mF, and E(t)=155sin(377t)V. Q(0)=I(0)=0.
Thus, I=Ih+ Ip
Apply the initial conditions
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82. 2nd Order Differential Equations
Applications:
Ex2:
Two cases of RLC Circuit
Find the current I(t) in the given RLC circuit with R=100Ω, L=0.1 H, C=1mF, and E(t)=155sin(377t)V. Q(0)=I(0)=0.
Applying the initial conditions
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83. 2nd Order Differential Equations
Applications:
1. Free falling stone
where s is distance or height and
g is acceleration due to gravity.
2. Spring vertical displacement
where y is displacement,
m is mass and
k is spring constant
3. RLC – circuit, Kirchoff ’s Second Law
q is charge on capacitor,
L is inductance,
c is capacitance.
R is resistance and
E is voltage
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84. 2nd Order Differential Equations
Applications:
4. Newton’s Low of Cooling
where
is rate of cooling of the liquid,
is temperature difference between the liquid ‘T’
and its surrounding Ts
5. Growth and Decay
y is the quantity present at any time
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85. Higher Order Differential Equations
The most general nth order linear differential equation is,
Thus, the standard form could be written as
In this type of D.Es. The roots/basis of the characteristics function becomes larger and the use of Wronskian plays a vital rule in the solution.
In this case the GENERAL SOLUTION (GS)will be extended to be in the form of:
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86. Higher Order Differential Equations
The basis of the GS are linearly independent if and only if the Wronskian (the n-th order determinant )is not equals to zero
In this case the Wronskian is given by
on some point in an open interval
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87. Higher Order Differential Equations
As previously explained for
n-th order D.E.
r(x)=0
Homogenous
r(x)0
Inhomogeneous
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88. Higher Order Differential Equations
Homogenous Higher O.D.Es with Constant coefficients
The solution will be based on the same solution method of the 2nd O.D.Es. (substitute )
Thus, there will be three different cases
Real and Unequal Roots
Complex Roots
Repeated Roots
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89. Higher Order Differential Equations
Real and Unequal Roots
Homogenous Higher O.D.Es with Constant coefficients
If roots of characteristic polynomial P(m) are real and unequal, then there are n distinct solutions of the differential equation:
If these functions are linearly independent, then general solution of differential equation is
The Wronskian can be used to determine linear independence of solutions.
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90. Higher Order Differential Equations
Complex Roots
Homogenous Higher O.D.Es with Constant coefficients
If the characteristic polynomial P(r) has complex roots, then they must occur in conjugate pairs,
Note that: not all the roots needed be complex.
Thus, General Solution
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91. Higher Order Differential Equations
Repeated Roots
Homogenous Higher O.D.Es with Constant coefficients
Suppose a root m of characteristic polynomial P(r) is a repeated root with multiplicity n. Then linearly independent solutions corresponding to this repeated root have the form
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92. Use Long Division (Based on the basis of 2)
Higher Order Differential Equations
Homogenous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
The auxiliary equation is
Use Long Division (Based on the basis of 2)
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93. Higher Order Differential Equations
Homogenous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
Thus the CF is
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94. Higher Order Differential Equations
Homogenous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
The auxiliary equation is
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95. Higher Order Differential Equations
Homogenous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
Thus the CF is
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96. Higher Order Differential Equations
Homogenous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
The auxiliary equation is
Thus the CF is
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97. Higher Order Differential Equations
Homogenous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
The auxiliary equation is
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98. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es with Constant coefficients
The Standard form is as follows
The Constant-coefficient equation
General Solution
Constants
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99. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
The auxiliary equation is
Thus the CF is
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100. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
Finding the PI: (Back to slide 45)
If we take it, we will see that its been contained in the homogenous solution
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101. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
Finding the PI: (Back to slide 45)
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102. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
Get the first and second derivatives of the general solution
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103. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
Substitute the initial conditions
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104. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
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105. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
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106. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es with Constant coefficients
Example: Solve the following D.E.
Solution:
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107. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es
Method of variation of parameters.
The standard form of the general solution is given by
where
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108. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es
Example: Solve the following D.E.
Solution:
Euler-Cauchy Equation
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109. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es
Example: Solve the following D.E.
Solution:
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110. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es
Example: Solve the following D.E.
Solution:
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111. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es
Example: Solve the following D.E.
Solution:
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112. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es
Example: Solve the following D.E.
Solution:
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113. Higher Order Differential Equations
Inhomogeneous Higher O.D.Es
Example: Solve the following D.E.
Solution:
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