S. Awad, Ph.D.

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Differential Equations
Math Review with Matlab
First Order Constant Coefficient Linear
Differential Equations

• S. Awad, Ph.D.
• M. Corless, M.S.E.E.
• E.C.E. Department
• University of Michigan-Dearborn

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First Order Constant Coefficient Linear
Differential Equations

• First Order Differential Equations
• General Solution of a First Order Constant
  Coefficient Differential Equation
• Electrical Applications
• RC Application Example

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First Order D.E.

• A General First Order Linear Constant Coefficient
  Differential Equation of x(t) has the form

• Where a is a constant and the function f(t) is
  given

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Properties

• A General First Order Linear Constant Coefficient
  DE of x(t) has the properties

• The DE is a linear combination of x(t) and its
  derivative
• x(t) and its derivative are multiplied by
  constants

• There are no cross products

• In general the coefficient of dx/dt is normalized
  to 1

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Fundamental Theorem

• A fundamental theorem of differential equations
  states that given a differential equation of the
  form below where x(t)xp(t) is any solution to

SOLUTION

• and x(t)xc(t) is any solution to the homogenous
  equation

SOLUTION

• Then x(t) xp(t)xc(t) is also a solution to the
  original DE

SOLUTION
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f(t) Constant Solution

• If f(t) b (some constant) the general solution
  to the differential equation consists of two
  parts that are obtained by solving the two
  equations

xp(t) Particular Integral Solution
xc(t) Complementary Solution
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Particular Integral Solution

• Since the right-hand side is a constant, it is
  reasonable to assume that xp(t) must also be a
  constant

• Substituting yields

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Complementary Solution

• To solve for xc(t) rearrange terms

• Which is equivalent to

• Integrating both sides

• Taking the exponential of both sides

• Resulting in

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First Order Solution Summary

• A General First-Order Constant Coefficient
  Differential Equation of the form

a and b are constants

• Has a General Solution of the form

K1 and K2 are constants
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Particular and Complementary Solutions
Particular Integral Solution
Complementary Solution
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Determining K1 and K2

• In certain applications it may be possible to
  directly determine the constants K1 and K2

• The first relationship can be seen by evaluating
  for t0

• The second by taking the limit as t approaches
  infinity

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Solution Summary

• By rearranging terms, we see that given
  particular conditions, the solution to

a and b are constants

• Takes the form

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Electrical Applications

• Basic electrical elements such as resistors (R),
  capacitors (C), and inductors (L) are defined by
  their voltage and current relationships

• A Resistor has a linear relationship between
  voltage and current governed by Ohms Law

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Capacitors and Inductors

• A first-order differential equation is used to
  describe electrical circuits containing a single
  memory storage elements like a capacitors or
  inductor

• The current and voltage relationship for a
  capacitor C is given by

• The current and voltage relationship for an
  inductor L is given by

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RC Application Example

• Example For the circuit below, determine an
  equation for the voltage across the capacitor for
  tgt0. Assume that the capacitor is initially
  discharged and the switch closes at time t0

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Plan of Attack

• Write a first-order differential equation for the
  circuit for time tgt0
• The solution will be of the form K1K2e-at
• These constants can be found by
• Determining a
• Determining vc(0)
• Determining vc()
• Finally graph the resulting vc(t)

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Equation for t gt 0

• Kirchhoffs Voltage Law (KVL) states that the sum
  of the voltages around a closed loop must equal
  zero

• Ohms Law states that the voltage across a
  resistor is directly proportional to the current
  through it, VIR

• Use KVL and Ohms Law to write an equation
  describing the circuit after the switch closes

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Differential Equation

• Since we want to solve for vc(t), write the
  differential equation for the circuit in terms of
  vc(t)

• Replace i Cdv/dt for capacitor current voltage
  relationship

• Rearrange terms to put DE in Standard Form

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General Solution

• The solution will now take the standard form

• a can be directly determined

• K1 and K2 depend on vc(0) and vc()

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Initial Condition

• A physical property of a capacitor is that
  voltage cannot change instantaneously across it

• Therefore voltage is a continuous function of
  time and the limit as t approaches 0 from the
  right vc(0-) is the same as t approaching from
  the left vc(0)

• Before the switch closes, the capacitor was
  initially discharged, therefore

• Substituting gives

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Steady State Condition

• As t approaches infinity, the capacitor will
  fully charge to the source VDC voltage

• No current will flow in the circuit because there
  will be no potential difference across the
  resistor, vR() 0 V

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Solve Differential Equation

• Now solve for K1 and K2

• Replace to solve differential equation for vc(t)

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Time Constant

• When analyzing electrical circuits the constant
  1/a is called the Time Constant t

K1 Steady State Solution
t Time Constant

• The time constant determines the rate at which
  the decaying exponential goes to zero

• Hence the time constant determines how long it
  takes to reach the steady state constant value of
  K1

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Plot Capacitor Voltage

• For First-order RC circuits the Time Constant t
  1/RC

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Summary

• Discussed general form of a first order constant
  coefficient differential equation

• Proved general solution to a first order constant
  coefficient differential equation

• Applied general solution to analyze a resistor
  and capacitor electrical circuit