Differential Equations
for Engineers:
the Essentials
Class 3 notes
Agenda: Class 3
Reviewing Homework Assignment 1
Lectures:
First order linear differential equations
System viewpoint
First order nonlinear differential equations
Homework Assignment 3
First Order Differential Equations
Linear Equations:
The System Viewpoint
Air at temperature
tTTT
sin
0
+=
Review: Exposed Water Pipe in Cyclical
Ambient Temperature
Review of Water Pipe Temperature Problem
Recall from the last class that the governing ODE was
and its solution was
Review of Water Pipe Temperature Problem (2)
deT tt)sin(
0
)(
0−−
)( −− t
e
is called the “kernel” and
Note that the second term in Equation 1 can be written
Review of Water Pipe Temperature Problem (3)
The convolution and the kernel are important concepts in the
analysis of linear differential equations of all types. The general
form for linear time-invariant systems is
−= tdgtKty 0)()()(
Review of Water Pipe Temperature Problem (4)
Defining
22
cos
+
=
22
sin
+
=
Solution to Water Pipe Temperature Problem (5)
The second term in Equation 2 is a transient.
The coefficient of
the amplitude
)sin(
0
−tT
,
The “System”
To engineers, the system is whatever the differential equation is
modeling.
Air temperature
tTTT
sin
0
+=
d
Wall thickness =
System Example: Water Pipe in Cyclical
Ambient Temperature
Input:
Block Diagram
I
T
System Stability
A linear system is said to be stable if its response to bounded initial
conditions is bounded.
A linear time-invariant system is damped if its response to bounded
initial conditions goes to zero as time goes to infinity.
First Order Differential Equations
Solving Nonlinear Separable Equations
Solving “separable” first order ODEs
tgyf
dt
dy
)()(
=
A first order ODE is said to be “separable” if it has the form
Sometimes one can integrate both sides:
=
)(
)(
yF
yf
dy
which can be written
Solving “separable” first order ODEs (2)
ctGyF += )()(
and then there results an implicit equation for yin terms of t :
Occasionally one can solve Equation 1 and find yas a function of t
Equation 1
In-class Problems
tbya
dt
dy
++=
))((
Solve explicitly:
First Order Differential Equations
Example: Sounding rocket*