Differential Equations EXACT EQUATIONS
[Pages:34]Differential Equations
EXACT EQUATIONS
Graham S McDonald A Tutorial Module for learning the technique
of solving exact differential equations
q Table of contents q Begin Tutorial
c 2004 g.s.mcdonald@salford.ac.uk
Table of contents
1. Theory 2. Exercises 3. Answers 4. Standard integrals 5. Tips on using solutions
Full worked solutions
Section 1: Theory
3
1. Theory
We consider here the following standard form of ordinary differential equation (o.d.e.):
P (x, y)dx + Q(x, y)dy = 0
If
P y
=
Q x
then
the
o.de.
is
said
to
be
exact.
This means that a function u(x, y) exists such that:
u
u
du = dx + dy
x
y
= P dx + Q dy = 0 .
One
solves
u x
=
P
and
u y
=
Q
to
find
u(x, y).
Then du = 0 gives u(x, y) = C, where C is a constant. This last equation gives the general solution of P dx + Q dy = 0.
Toc
Back
Section 2: Exercises
4
2. Exercises
Click on Exercise links for full worked solutions (there are 11 exercises in total)
Show that each of the following differential equations is exact and use that property to find the general solution:
Exercise 1.
1
y
dy x
-
x2
dx
=
0
Exercise 2.
dy 2xy
+
y2
-
2x
=
0
dx
Exercise 3. 2(y + 1)exdx + 2(ex - 2y)dy = 0
q Theory q Answers q Integrals q Tips
Toc
Back
Section 2: Exercises
5
Exercise 4. (2xy + 6x)dx + (x2 + 4y3)dy = 0
Exercise 5. (8y - x2y) dy + x - xy2 = 0
dx
Exercise 6. (e4x + 2xy2)dx + (cos y + 2x2y)dy = 0
Exercise 7. (3x2 + y cos x)dx + (sin x - 4y3)dy = 0
q Theory q Answers q Integrals q Tips
Toc
Back
Section 2: Exercises
6
Exercise 8.
x
tan-1
y
?
dx
+
x2 2(1 +
y2)
?
dy
=
0
Exercise 9. (2x + x2y3)dx + (x3y2 + 4y3)dy = 0
Exercise 10. (2x3 - 3x2y + y3) dy = 2x3 - 6x2y + 3xy2
dx
Exercise 11. (y2 cos x - sin x)dx + (2y sin x + 2)dy = 0
q Theory q Answers q Integrals q Tips
Toc
Back
Section 3: Answers
3. Answers
1. y = Ax ,
2. y2x - x2 = A , 3. (y + 1)ex - y2 = A , 4. x2y + 3x2 + y4 = A ,
5.
1 2
x2(1
-
y2)
+
4y2
=
A
,
6.
1 4
e4x
+
x2y2
+
sin
y
=
A
,
7. x3 + y sin x - y4 = A ,
8.
x2 2
tan-1
y
=
A
,
9.
x2
+
x3 y 3 3
+ y4 = A ,
Toc
7 Back
Section 3: Answers
8
10.
x4 2
- 2x3y +
3 2
x2y2
-
y4 4
=A,
11. y2 sin x + cos x + 2y = A.
Toc
Back
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- lx y3ft y l 1o llld university of utah
- solutions section 2 whitman college
- problem 1 find the point on the plane 4x 3y z 10 nearest to 2 0 1
- special right triangles practice lcps
- pre ap algebra 2 lesson 2 2 determining the number of denton isd
- x 2y 6
- p example 7 8 0 p solution d p michigan state university
- mathematics 205 section 16 3 p755 wellesley college
- chamblee middle school
- math 263 assignment 9 solutions solution z z