Differential Equations
for Engineers:
the Essentials
Class 4 notes
Agenda: Class 4
Review homework assignment 2
Lectures:
Homework assignment 4
Short Quiz
First Order Differential Equations
Definition of Terms
=
=
=
m
V
h
Rocket altitude
Rocket vertical velocity
Rocket mass
Sounding rocket: Summarizing analysis of
phase one
Phase 1: High-thrust segment
Starting with
we eliminated time from the equation and found
from which we derived
Sounding rocket: Summarizing analysis of
phase two
b
Phase 2: No-thrust segment going up
Starting with
we found that the maximum altitude (altitude at apogee) is
No-thrust segment coming down
Objective of Phase 3 analysis: determine the maximum speed the
sounding rocket achieves on its way down
Again, Newton’s second law: Force = mass times acceleration.
Now force = weight downward and drag upward:
No-thrust segment coming down (2)
This is another example of a “separable” first order DE.
We make the same simplifying substitution as before:
No-thrust segment coming down (3)
We solve this with partial fraction expansion.
Using equation 1 and factoring the denominator we can write:
We find the values of the constants by recombining:
No-thrust segment coming down (4)
Then Equation 2 becomes
Integrating both sides, as time goes from to
and the (scaled) velocity goes from zero to
a
t
v
t
No-thrust segment coming down (5)
For simplicity let
Then equation 3 becomes
No-thrust segment coming down (6)
the vertical velocity after apogee is
where
No-thrust segment coming down (7)
Remember that, in this third phase, we are interested in finding the
maximum speed of the rocket on the way down. One can easily verify
Approach: integrate Equation 5 to obtain , find the time when the
Challenge Problem CP3.2 invites you to use this approach to show that
)(th
t
First Order Differential Equations
Possibility of transforming nonlinear into
linear
Transforming nonlinear into linear
There is a shortcut to solving the problems we have been considering.
From the point of view of a student just learning elementary
methods of solving ordinary differential equations it is essentially a
trick.
Transforming nonlinear into linear (2)
Let us reconsider Phase 2 of our analysis, the no-thrust region on the
way up. In Phase 2 the equation of motion was
We transform from the independent variable tto the independent
variable h(altitude). Then
Transforming nonlinear into linear (3)
Now we transform from the dependent variable Vto the dependent
Then using Equations 6, 7 and 8 we arrive at
We have transformed a nonlinear equation into a linear one.
Transforming nonlinear into linear (4)
Word Problem WP3.1 asks you to use the shortcut method just
outlined to show that
(a) the rocket’s highest altitude is
(b) the maximum speed the rocket reaches on the way down is
How to Check
as we expect. (From conservation of energy, in absence of drag, the
velocity at a given altitude is the same coming down as going up.)
For very large k, the equation becomes
Homework Assignment 4
Read:
Chapter 3, Sections 3.2 and 3.3