Lecture 19
●
MEEN 357 Homework Solution
1
Lecture19HomeworkSolution.doc
RMB
1.
Consider the following nonlinear first order ordinary differential equation
( )
3
5
1
5
where
0
2
20
dx
x
tx
x
dt
−
= −
=
(19.1)
a.
This particular ordinary differential equation is one studied in Math 308.
It is known as
a Bernoulli equation.
With this hint, show that the exact solution is
( )
10
10
50
5
2005
t
x t
t
e
−
=
−
+
(19.2)
Solution
:
You can solve this problem with dsolve and that is acceptable.
It is also easily solved by
hand.
If you check out the solution method for Bernoulli equations, you will find that it is a method
that works for ordinary differential equations of the form
( )
( )
( )
( )
n
dx t
P t x t
Q t x
dt
+
=
(19.3)
This equation solved by first changing variables with the substitution
1
1
n
y
x
−
=
(19.4)
In the case of (19.1), the substitution is
2
1
y
x
=
(19.5)
Given this result, it perhaps becomes evident that (19.1) can be written
2
2
1
1
1
5
5
2
2
d
x
t
dt
x
−
−
= −
(19.6)
or, by simple rearrangement,
2
2
1
1
10
5
d
x
t
dt
x
+
=
(19.7)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Lecture 19
●
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 ANAMALAI

Click to edit the document details