1. Differential Equations
EGR 1101: Unit 12 Lecture #1
Differential Equations
(Sections 10.1 to 10.4 of Rattan/Klingbeil text)
Fire up MATLAB

2. 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.

3. Some Examples
Examples of linear ordinary differential equation with constant coefficients:

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.
Note forcing function in 3 examples on previous slide.

5. 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.

6. A Procedure for Solving Linear ODEs with Constant Coefficients
We’ll use a four-step procedure for solving this type of equation:
Find the transient solution.
Find the steady-state solution.
Find the total solution by adding the results of Steps 1 and 2.
Apply initial conditions (if given) to evaluate unknown constants that arose in the previous steps.
See pages in Rattan/Klingbeil textbook.

7. 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!

8. Some Equations that Our Procedure Can’t Handle
Nonlinear differential equations
Partial differential equations
Diff eqs whose coefficients depend on y or t
In later courses you’ll learn procedures for dealing with these.

9. MATLAB Commands Without initial conditions:
>>dsolve('2*Dy + y = 8')
With initial conditions:
>>dsolve('2*Dy + y = 8', 'y(0)=5')

10. 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')

11. Today’s Examples
Leaking bucket with constant inflow rate and bucket initially empty
Leaking bucket with zero inflow and bucket initially filled to a given level

12. First-Order Differential Equations in Electrical Systems
EGR 1101: Unit 12 Lecture #2
First-Order Differential Equations in Electrical Systems
(Section 10.4 of Rattan/Klingbeil text)

13. Review: Procedure
Steps in solving a linear ordinary differential equation with constant coefficients:
Find the transient solution.
Find the steady-state solution.
Find the total solution by adding the results of Steps 1 and 2.
Apply initial conditions (if given) to evaluate unknown constants that arose in the previous steps.

14. 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.

15. Today’s Examples Series RC circuit with constant source voltage
First-order low-pass filter

16. Exponentially Saturating Function
A function of the form
𝑓 𝑡 =𝐾 (1−𝑒 −𝑡/𝜏 )
where K and  are constants, is called an exponentially saturating function.
At t = 0, f(t) = 0.
As t  , f(t)  K.

17. Exponentially Saturating Function: Time Constant
In 𝑓 𝑡 =𝐾 (1−𝑒 −𝑡/𝜏 ), 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.

18. Time Constant Rules of Thumb
For 𝑓 𝑡 =𝐾 (1−𝑒 −𝑡/𝜏 ),
When t = , f(t)   K. (After one time constant, the function has risen to about 63.2% of its limiting value.)
When t = 5  , f(t)   K. (After five time constants, the function has risen to about 99.3% of its limiting value.)
See next slide for graph.

19. Exponentially Saturating Function: Graph

20. Exponentially Decaying Function
A function of the form
𝑓 𝑡 =𝐾 𝑒 −𝑡/𝜏
where K and  are constants, is called an exponentially decaying function.
At t = 0, f(t) = K.
As t  , f(t)  0.

21. Exponentially Decaying Function: Time Constant
In 𝑓 𝑡 =𝐾 𝑒 −𝑡/𝜏 , 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.

22. Time Constant Rules of Thumb
For 𝑓 𝑡 =𝐾 𝑒 −𝑡/𝜏 ,
When t = , f(t)   K. (After one time constant, the function has fallen to about 36.8% of its initial value.)
When t = 5  , f(t)   K. (After five time constants, the function has fallen to about 0.7% of its initial value.)
See next slide for graph.

23. Exponentially Decaying Function: Graph

24. 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.