Differential Equations HOMOGENEOUS FUNCTIONS
Differential Equations
HOMOGENEOUS FUNCTIONS
Graham S McDonald A Tutorial Module for learning to solve
differential equations that involve homogeneous functions
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
M (x, y) = 3x2 + xy is a homogeneous function since the sum of the powers of x and y in each term is the same (i.e. x2 is x to power 2 and xy = x1y1 giving total power of 1 + 1 = 2).
The degree of this homogeneous function is 2.
Here, we consider differential equations with the following standard form:
dy M (x, y) =
dx N (x, y)
where M and N are homogeneous functions of the same degree.
Toc
Back
Section 1: Theory
4
To find the solution, change the dependent variable from y to v, where
y = vx .
The LHS of the equation becomes: dy dv =x +v dx dx
using the product rule for differentiation.
Solve the resulting equation by separating the variables v and x. Finally, re-express the solution in terms of x and y.
Note. This method also works for equations of the form:
dy
y
=f
.
dx
x
Toc
Back
Section 2: Exercises
5
2. Exercises
Click on Exercise links for full worked solutions (there are 11 exercises in total)
Exercise 1.
dy xy + y2
Find the general solution of = dx
x2
Exercise 2.
Solve
dy 2xy
=
x2
+ y2
given
that
y
=
0
at
x
=
1
dx
Exercise 3.
dy x + y
Solve =
and find the particular solution when y(1) = 1
dx x
q Theory q Answers q Integrals q Tips
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
- differential equations homogeneous functions
- sample problems
- sample problems aceh
- 7 4 proving trigonometric identities a equations versus
- ecuaciones trigonométricas resueltas
- chapter 3 7 derivatives of the trigonometric functions
- trigonometry identities i introduction
- 18 verifying trigonometric identities
- section 7 2 advanced integration techniques trigonometric
- trigonometry laws and identities