EGR 1101: Unit 12 Lecture #1

Differential Equations

(Sections 10.1 to 10.4 of Rattan/Klingbeil text)

Linear ODE with Constant Coefficients

Given independent variable t and dependent variable y(t), a linear ordinary differential equation with constant coefficients is an equation of the form

where A0, A1, …, An, are constants.

)()(... 01 tftyAdtdyA

dtydA n

n

n

Some Examples

Examples of linear ordinary differential equation with constant coefficients:

82 ydtdy

tydtdy

dtyd 3652

2

tdtdy

dtyd

dtyd sin83 3

3

4

4

Forcing Function

In the equation

the function f(t) is called the forcing function.

It can be a constant (including 0) or a function of t, but it cannot be a function of y.

)()(... 01 tftyAdtdyA

dtydA n

n

n

Solving Linear ODEs with Constant Coefficients

Solving one of these equations means finding a function y(t) that satisfies the equation.

• You already know how to solve some of these equations, such as

• But many equations are more complicated and cannot be solved just by integrating.

2dtdy

A Procedure for Solving Linear ODEs with Constant Coefficients We’ll use a four-step procedure for solving

this type of equation:1. Find the transient solution.2. Find the steady-state solution.3. Find the total solution by adding the results of

Steps 1 and 2.4. Apply initial conditions (if given) to evaluate

unknown constants that arose in the previous steps.

See pages 371-372 in Rattan/Klingbeil textbook.

Forcing Function = 0?

If the forcing function (the right-hand side of your differential equation) is equal to 0, then the steady-state solution is also 0.

In such cases, you get to skip straight from Step 1 to Step 3!

Some Equations that Our Procedure Can’t Handle

Nonlinear differential equations

Partial differential equations

Diff eqs whose coefficients depend on y or t

tydtdy sin)7(2 3

2

272 txy

ty

072 ydtdyt

MATLAB Commands

Without initial conditions:

>>dsolve('2*Dy + y = 8')

With initial conditions:

>>dsolve('2*Dy + y = 8', 'y(0)=5')

MATLAB Commands

Without initial conditions:

>>dsolve('D2y+5*Dy+6*y=3*t')

With initial conditions:

>>dsolve('D2y+5*Dy+6*y=3*t', 'y(0)=0', 'Dy(0)=0')

Today’s Examples

1. Leaking bucket with constant inflow rate and bucket initially empty

2. Leaking bucket with zero inflow and bucket initially filled to a given level

EGR 1101: Unit 12 Lecture #2

First-Order Differential Equations in Electrical Systems

(Section 10.4 of Rattan/Klingbeil text)

Review: Procedure

Steps in solving a linear ordinary differential equation with constant coefficients:

1. Find the transient solution.2. Find the steady-state solution.3. Find the total solution by adding the results of

Steps 1 and 2.4. Apply initial conditions (if given) to evaluate

unknown constants that arose in the previous steps.

Forcing Function = 0?

Recall that if the forcing function (the right-hand side of your differential equation) is equal to 0, then the steady-state solution is also 0.

In such cases, you get to skip straight from Step 1 to Step 3.

Today’s Examples

1. Series RC circuit with constant source voltage

2. First-order low-pass filter

Exponentially Saturating Function

A function of the form

where K and are constants, is called an exponentially saturating function. At t = 0, f(t) = 0. As t , f(t) K.

Exponentially Saturating Function: Time Constant

In , the quantity is called the time constant.

The time constant is a measure of how quickly or slowly the function rises.

The greater is, the more slowly the function approaches its limiting value K.

Time Constant Rules of Thumb

For , When t = , f(t) 0.632 K.

(After one time constant, the function has risen to about 63.2% of its limiting value.)

When t = 5 , f(t) 0.993 K. (After five time constants, the function has risen to about 99.3% of its limiting value.)

See next slide for graph.

Exponentially Saturating Function: Graph

Exponentially Decaying Function

A function of the form

where K and are constants, is called an exponentially decaying function. At t = 0, f(t) = K. As t , f(t) 0.

Exponentially Decaying Function: Time Constant

In , the quantity is called the time constant.

The time constant is a measure of how quickly or slowly the function falls.

The greater is, the more slowly the function approaches 0.

Time Constant Rules of Thumb

For , When t = , f(t) 0.368 K.

(After one time constant, the function has fallen to about 36.8% of its initial value.)

When t = 5 , f(t) 0.007 K. (After five time constants, the function has fallen to about 0.7% of its initial value.)

See next slide for graph.

Exponentially Decaying Function: Graph

Low-Pass and High-Pass Filters

A low-pass filter is a circuit that passes low-frequency signals and blocks high-frequency signals.

A high-pass filter is a circuit that does just the opposite: it blocks low-frequency signals and passes high-frequency signals.