1. Differential Equations and its Applications in Engineering
Prof. (Dr.) Nita H. Shah
Department of Mathematics,
Gujarat University, Ahmedabad

2. Modeling with differential equations
One of the most important application of calculus is
differential equations, which often arise in describing
some phenomenon in engineering, physical science
etc.
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3. Concepts of differential equations
In general, a differential equation is an equation that
contains an unknown function and its derivatives. The order
of a differential equation is the order of the highest derivative
that occurs in the equation.
A function y=f(x) is called a solution of a differential
equation if the equation is satisfied when y=f(x) and its
derivatives are substituted into the equation.
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4. 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
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5. 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
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6. 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
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7. 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
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8. 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.
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9. 3. 4. is non – linear Example: Example:
is non - linear because in 2nd term coefficient depends on y. 3.
Example: 4.
is non - linear because
is non – linear
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10. It is Ordinary/partial Differential equation of order… and of degree…, it is linear / non linear, with independent variable…, and dependent variable….
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11. 1st – order differential equation
1. Derivative form:
2. Differential form: .
3. General form: or
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Classfication of Differential Equation

12. First Order Ordinary Differential equation
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13. Second order Ordinary Differential Equation
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Classfication of Differential Equation

14. nth – order linear differential equation
1. nth – order linear differential equation with constant coefficients.
2. nth – order linear differential equation with variable coefficients
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15. Solution of Differential Equation
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16. Examples
y=2x+c , is solution of the 1st order differential equation , c is arbitrary constant.
As is solution of the differential equation for every value of c,
hence it is known as general solution.
Examples
1 parameter
2 parameters
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17. Families of Solutions Example Solution
Observe that given any point (x0,y0), there is a unique solution curve of the above
equation which curve goes through the
given point.
The solution is a family of ellipses.

18. Origin of differential equations solution
Geometric Origin
1. For the family of straight lines
the differential equation is
2. For the family of curves A.
the differential equation is . B.
the differential equation is
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19. g is acceleration due to gravity.
Physical Origin
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
where q is charge on capacitor, L is inductance, c is
capacitance, R is resistance and E is voltage
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20. Physical Origin Newton’s Low of Cooling where and its surrounding Ts
is rate of cooling of the liquid,
is temperature difference between the liquid ‘T’
and its surrounding Ts
2. Growth and Decay
where y is the quantity present at any time
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21. Geometric point of view
Geometrically, the general solution is a family of solution curves, which are called integral curves.
When we impose an initial condition, we look at the family of
solution curves and pick the one that passes through the point
Physically, this corresponds to measuring the state of a system at time and using the solution of the initial-value problem to predict the future behavior of the system.
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22. Graphical approach: direction fields
For most differential equations, it is impossible to find an explicit formula for the solution.
Suppose we are asked to sketch the graph of the solution of the initial-value problem
The equation tells us that the slope at any point (x,y) on the graph is f(x,y).
To sketch the solution curve, we draw short line segments with slope f(x,y) at a number of points (x,y). The result is
called a direction field.
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23. Example
Ex. Draw a direction for the equation What can you say about the limiting value when
Sol.
Remark: equilibrium solution
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24. Separable equations
Not all equations have an explicit formula for a solution.
But some types of equations can be solved explicitly.
Among others, separable equations is one type.
A separable equation is a first-order differential equation in
which the expression for can be factored into the
product of a function of x and a function of y. That is, a
separable equation can be written in the form
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25. Solutions of separable equations
Thus a separable equation can be written into
that is, the variables x and y are separated.
We can then integrate both sides to get
After we find the indefinite integrals, we get a relationship
between x and y, in which there generally has an arbitrary
constant. So the relationship determines a function y = y(x) and
it is the general solution to the differential equation.
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26. Example Ex. Solve the differential equation
Sol. Rewrite the equation into
Integrate both sides which gives
So, the general solution is
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27. Example Ex. Solve the differential equation Sol. Separate variables:
Integrate:
which is the general solution in implicit form.
Remark: It is impossible to solve y in terms of x explicitly.
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28. Example Ex. Solve the differential equation Sol.
C is arbitrary, but is not arbitrary.
Verify that y=0 is also a solution. Therefore,
where A is an arbitrary constant, is the general solution.
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29. Orthogonal trajectories
An orthogonal trajectory of a family of curve is a curve
that intersects each curve of the family orthogonally. For
instance, each member of the family of straight lines
is an orthogonal trajectory of the family
To find orthogonal trajectories of a family of curve, first
find the slope at any point on the family of curve, which is
generally a differential equation. At any point on the
orthogonal trajectories, the slope must be the negative
reciprocal of the aforementioned slope. So the slope of
orthogonal trajectories is governed by a differential equation,
too. Let us solve the equation to get the orthogonal trajectories.
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30. Example Ex. Find the orthogonal trajectories of the family of curves
where k is an arbitrary constant.
Sol. Separating variables gives or
Substituting into it, we find the slope at any point is
At any point on orthogonal trajectory, the slope is
Solving the equation, we get
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31. Example Ex. Suppose f is continuous and Find f(x). Sol.
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32. First order differential equations
No general method of solutions of 1st O.D.E.s because of their different degrees of complexity.
Possible to classify them as:
exact equations
equations in which the variables can be separated
homogenous equations
equations solvable by an integrating factor
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33. Exact equations Exact? General solution: F (x,y) = C For example
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34. Separable-variables equations
In the most simple first order differential eqs., the independent variable and its differential can be
separated from the dependent variable and its
differential by the equality sign, using nothing
more than the normal processes of elementary
algebra.
For example
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35. Homogeneous equations
Homogeneous/nearly homogeneous?
A differential equation of the type,
Such an equation can be solved by making the substitution u = y/x and thereafter integrating the transformed equation.
is termed a homogeneous differential equation of
the first order.
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36. Homogeneous equation example
Liquid benzene is to be chlorinated batchwise by sparging chlorine gas into a reaction kettle containing the benzene. If the reactor
contains such an efficient agitator that all the chlorine which
enters the reactor undergoes chemical reaction, and only the
hydrogen chloride gas liberated escapes from the vessel, estimate
how much chlorine must be added to give the maximum yield of
monochlorbenzene. The reaction is assumed to take place isothermally at 55 C when the ratios of the specific reaction rate constants are:
C6H6+Cl2  C6H5Cl +HCl
C6H5Cl+Cl2  C6H4Cl2 + HCl
C6H4Cl2 + Cl2  C6H3Cl3 + HCl
k1 = 8 k2 ; k2 = 30 k3
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37. Take a basis of 1 mole of benzene fed to the reactor and introduce
the following variables to represent the stage of system at time ,
p = moles of chlorine present
q = moles of benzene present
r = moles of monochlorbenzene present
s = moles of dichlorbenzene present
t = moles of trichlorbenzene present
Then q + r + s + t = 1
and the total amount of chlorine consumed is: y = r + 2s + 3t
From the material balances : in - out = accumulation
u = r/q
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38. Equations solved by integrating factor
There exists a factor by which the equation can be multiplied so that the one side becomes a complete differential equation. The factor is called “the integrating factor”.
where P and Q are functions of x only
Assuming the integrating factor R is a function of x only, then
is the integrating factor.
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39. Example
Solve
Let z = 1/y3
integral factor
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40. First order linear differential equations
Summary of 1st O.D.E.
First order linear differential equations
occasionally arise in chemical engineering
problems in the field of heat transfer,
momentum transfer and mass transfer.
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41. First O.D.E. in heat transfer
An elevated horizontal cylindrical tank 1 m diameter and 2 m long is
insulated with asbestos lagging of thickness l = 4 cm, and is employed
as a maturing vessel for a batch chemical process. Liquid at 950 C is
charged into the tank and allowed to mature over 5 days.
If the data below applies, calculated the final temperature of the
liquid and give a plot of the liquid temperature as a function of time.
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42. Liquid film coefficient of heat transfer (h1) = 150 W/m2C
Thermal conductivity of asbestos (k) = 0.2 W/m C
Surface coefficient of heat transfer by
convection and radiation (h2) = 10 W/m2C
Density of liquid () = kg/m3
Heat capacity of liquid (s) = 2500 J/kgC
Atmospheric temperature at time of charging = 20 C
Atmospheric temperature (t) t = cos (/12)
Time in hours ()
Heat loss through supports is negligible. The thermal
capacity of the lagging can be ignored.
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43. Area of tank (A) = ( x 1 x 2) + 2 ( 1 / 4  x 12 ) = 2.5 m2
Rate of heat loss by liquid = h1 A (T - Tw)
Rate of heat loss through lagging = kA/l (Tw - Ts)
Rate of heat loss from the exposed surface of the lagging = h2 A (Ts - t)
At steady state, the three rates are equal:
Considering the thermal equilibrium of the liquid,
input rate - output rate = accumulation rate t Ts Tw T
B.C.  = 0 , T = 95
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44. Second O.D.E. Purpose: reduce to 1st O.D.E.
Likely to be reduced equations:
Non-linear
Equations where the dependent variable does not occur
explicitly.
Equations where the independent variable does not occur
Homogeneous equations.
Linear
The coefficients in the equation are constant
The coefficients are functions of the independent variable.
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45. They are solved by differentiation followed by the p-substitution.
Non-linear 2nd O.D.E. - Equations where the dependent variables does not occur explicitly
They are solved by differentiation followed by the
p-substitution.
When the p-substitution is made, the
second derivative of y is replaced by the first
derivative of p thus eliminating y completely and
producing a first O.D.E. in p and x.
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46. Solve Let and therefore integral factor error function
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47. Non-linear 2nd O.D.E. - Equations where the independent variables does not occur explicitly
They are solved by differentiation followed by
the p-substitution.
When the p-substitution is made in this case,
the second derivative of y is replaced as
Let
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48. Solve Let and therefore Separating the variables
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49. Non-linear 2nd O.D.E.- Homogeneous equations
The homogeneous 1st O.D.E. was in the form:
The corresponding dimensionless group containing the 2nd
differential coefficient is
In general, the dimensionless group containing the nth
coefficient is
The second order homogenous differential equation can be
expressed in a form analogous to , viz.
Assuming u = y/x
Assuming x = et
If in this form, called homogeneous 2nd ODE
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50. Solve Dividing by 2xy homogeneous Let Let Singular solution
General solution
Let
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51. Linear differential equations
They are frequently encountered in most
chemical engineering fields of study, ranging from
heat, mass, and momentum transfer to applied
chemical reaction kinetics.
The general linear differential equation of the nth order having constant coefficients may be written:
where (x) is any function of x.
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52. 2nd order linear differential equations
The general equation can be expressed in the form
where P,Q, and R are constant coefficients
Let the dependent variable y be replaced by the sum of the two
new variables: y = u + v
If v is a particular solution of the original differential equation
The general solution of the linear differential equation will be the sum of
a “complementary function” and a “particular solution”.
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purpose

53. The complementary function
Let the solution assumed to be:
auxiliary equation (characteristic equation)
Unequal roots
Equal roots
Real roots
Complex roots
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54. Unequal roots to auxiliary equation
Let the roots of the auxiliary equation be distinct and of values
m1 and m2. Therefore, the solutions of the auxiliary equation are:
The most general solution will be
If m1 and m2 are complex it is customary to replace the complex exponential functions with their equivalent trigonometric forms.
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55. Solve
auxiliary function
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56. Equal roots to auxiliary equation
Let the roots of the auxiliary equation equal and of value m1 = m2 = m. Therefore, the solution of the auxiliary equation is:
Let
where V is a function of x
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57. Solve
auxiliary function
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58. Solve
auxiliary function
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59. Particular integrals
Two methods will be introduced to obtain the particular solution of a second linear O.D.E.
The method of undetermined coefficients
confined to linear equations with constant coefficients and particular form of  (x)
The method of inverse operators
general applicability
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60. Method of undetermined coefficients
When  (x) is constant, say C, a particular integral of eq.
When  (x) is a polynomial of the form where all the coefficients are constants. The form of a particular integral is
When  (x) is of the form Terx, where T and r are
constants. The form of a particular integral is
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61. Method of undetermined coefficients
When  (x) is of the form G sin nx + H cos nx, where G and H are constants, the form of a
particular solution is
Modified procedure when a term in the
particular integral duplicates a term in the
complementary function.
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62. Solve auxiliary equation Equating coefficients of equal powers of x
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63. Method of inverse operators
Sometimes, it is convenient to refer to the symbol “D” as the differential operator:
But,
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64. 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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65. The basic laws of algebra thus apply to the pure operators,
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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66. Differential operator to exponentials
More convenient!
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67. Differential operator to trigonometrical functions
where “Im” represents the imaginary part of the function which follows it.
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68. Parul Uni., Jan. 01, 2016