Date post: | 10-Apr-2018 |
Category: |
Documents |
Upload: | jose-villegas |
View: | 228 times |
Download: | 0 times |
of 36
8/8/2019 Ordinary Differential Equations Homogfn
1/36
Differential Equations
HOMOGENEOUS FUNCTIONS
Graham S McDonald
A Tutorial Module for learning to solvedifferential equations that involve
homogeneous functions
q Table of contents
q Begin Tutorial
c 2004 [email protected]
mailto:[email protected]://www.cse.salford.ac.uk/8/8/2019 Ordinary Differential Equations Homogfn
2/36
Table of contents
1. Theory
2. Exercises
3. Answers
4. Standard integrals
5. Tips on using solutions
Full worked solutions
8/8/2019 Ordinary Differential Equations Homogfn
3/36
Section 1: Theory 3
1. Theory
M(x, y) = 3x2
+ xy is a homogeneous function since the sum ofthe powers of x and y in each term is the same (i.e. x2 is x to power2 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 standardform:
dydx
= M(x, y)N(x, y)
where M and N are homogeneous functions of the same degree.
Toc Back
http://lastpage/http://prevpage/http://nextpage/http://nextpage/http://goback/http://goback/http://nextpage/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
4/36
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
dx= x
dv
dx+ v
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
dx = fy
x
.
Toc Back
http://lastpage/http://prevpage/http://nextpage/http://goback/http://goback/http://goback/http://goback/http://lastpage/http://prevpage/http://nextpage/http://goback/http://nextpage/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
5/36
Section 2: Exercises 5
2. Exercises
Click on Exercise links for full worked solutions (there are 11
exercises in total)
Exercise 1.
Find the general solution ofdy
dx
=xy + y2
x
2
Exercise 2.
Solve 2xydy
dx= x2 + y2 given that y = 0 at x = 1
Exercise 3.
Solvedy
dx=
x + y
xand find the particular solution when y(1) = 1
q Theory q Answers q Integrals q Tips
Toc Back
http://lastpage/http://prevpage/http://lastpage/http://lastpage/http://lastpage/http://lastpage/http://prevpage/http://lastpage/http://prevpage/http://prevpage/http://prevpage/http://nextpage/http://goback/http://nextpage/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
6/36
Section 2: Exercises 6
Exercise 4.
Solve xdy
dx
= x
y and find the particular solution when y(2) =
1
2
Exercise 5.
Solvedy
dx=
x 2yx
and find the particular solution when y(1) = 1
Exercise 6.
Given thatdy
dx=
x + y
x
y
, prove that tan1y
x
=
1
2ln
x2 + y2
+ A,
where A is an arbitrary constant
Exercise 7.
Find the general solution of 2x2dy
dx= x2 + y2
q Theory q Answers q Integrals q Tips
Toc Back
http://lastpage/http://prevpage/http://nextpage/http://nextpage/http://nextpage/http://goback/http://goback/http://lastpage/http://prevpage/http://prevpage/http://prevpage/http://nextpage/http://nextpage/http://nextpage/http://lastpage/http://lastpage/http://lastpage/http://goback/http://nextpage/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
7/36
Section 2: Exercises 7
Exercise 8.
Find the general solution of (2x
y)
dy
dx
= 2y
x
Note. The key to solving the next three equations is to
recognise that each equation can be written in the formdy
dx
= fyx f(v)
Exercise 9.
Find the general solution ofdy
dx=
y
x+ tan
yx
Exercise 10.
Find the general solution of xdy
dx= y + xe
y
x
q Theory q Answers q Integrals q Tips
Toc Back
http://lastpage/http://lastpage/http://lastpage/http://lastpage/http://lastpage/http://lastpage/http://prevpage/http://lastpage/http://prevpage/http://nextpage/http://goback/http://lastpage/http://lastpage/http://goback/http://nextpage/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
8/36
Section 2: Exercises 8
Exercise 11.
Find the general solution of xdy
dx
= y + x2 + y2
q Theory q Answers q Integrals q Tips
Toc Back
http://lastpage/http://goback/http://nextpage/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
9/36
Section 3: Answers 9
3. Answers
1. General solution is y = x
ln x+C ,
2. General solution is x = C(x2 y2) , and particular solution isx = x2 y2 ,
3. General solution is y = x ln (kx) , and particular solution isy = x + x ln x ,
4. General solution is 1 = Kx(x 2y) , and particular solution is2xy x2 = 2 ,
5. General solution is x2(x 3y) = K , and particular solution isx2(x 3y) = 4 ,
6. HINT: Try changing the variables from (x, y) to (x, v), where
y = vx ,
Toc Back
http://lastpage/http://lastpage/http://lastpage/http://lastpage/http://prevpage/http://nextpage/http://goback/http://lastpage/http://lastpage/http://prevpage/http://nextpage/http://goback/http://lastpage/http://lastpage/http://lastpage/http://lastpage/http://prevpage/http://nextpage/http://goback/http://lastpage/http://lastpage/http://lastpage/http://prevpage/http://nextpage/http://goback/http://lastpage/http://prevpage/http://prevpage/http://prevpage/http://nextpage/http://nextpage/http://nextpage/http://goback/http://goback/http://nextpage/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
10/36
Section 3: Answers 10
7. General solution is 2x = (x y)(ln x + C) ,
8. General solution is y x = K(x + y)3
,
9. General solution is sinyx
= kx ,
10. General solution is y = x ln( ln kx) ,
11. General solution is sinh1yx
= ln x + C .
Toc Back
http://lastpage/http://lastpage/http://lastpage/http://prevpage/http://prevpage/http://prevpage/http://lastpage/http://lastpage/http://lastpage/http://lastpage/http://lastpage/http://prevpage/http://lastpage/http://lastpage/http://lastpage/http://lastpage/http://prevpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
11/36
Section 4: Standard integrals 11
4. Standard integrals
f(x)
f(x)dx f(x)
f(x)dxxn x
n+1
n+1 (n = 1) [g (x)]n g (x) [g(x)]n+1
n+1 (n = 1)1x
ln |x| g(x)g(x) ln |g (x)|
ex ex ax ax
lna (a > 0)
sin x cos x sinh x cosh xcos x sin x cosh x sinh xtan x ln |cos x| tanh x ln cosh xcosec x ln
tan x2
cosech x lntanh x2
sec x ln |
sec x + tan x
|sech x 2tan1 ex
sec2 x tan x sech2 x tanh xcot x ln |sin x| coth x ln |sinh x|sin2 x x2 sin 2x4 sinh2 x sinh 2x4 x2cos2 x x2 +
sin 2x4 cosh
2 x sinh 2x4 +x2
Toc Back
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
12/36
Section 4: Standard integrals 12
f(x)
f(x) dx f(x)
f(x) dx
1
a2+x21
a tan1 x
a
1
a2x21
2a lna+xax (0 < |x|< a)
(a > 0) 1x2a2
12a ln
xax+a (|x| > a > 0)
1a2x2 sin
1 xa
1a2+x2 ln
x+a2+x2a
(a > 0)
(a < x < a) 1x2a2 ln
x+x2a2a (x > a > 0)
a2 x2 a22
sin1xa
a2 +x2 a22 sinh1 xa + xa2+x2a2
+xa2x2a2
x2a2 a22
cosh1 x
a
+ x
x2a2a2
Toc Back
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
13/36
Section 5: Tips on using solutions 13
5. Tips on using solutions
q When looking at the THEORY, ANSWERS, INTEGRALS orTIPS pages, use the Back button (at the bottom of the page) toreturn to the exercises.
q Use the solutions intelligently. For example, they can help you get
started on an exercise, or they can allow you to check whether yourintermediate results are correct.
q Try to make less use of the full solutions as you work your waythrough the Tutorial.
Toc Back
http://lastpage/http://lastpage/http://lastpage/http://prevpage/http://goback/http://lastpage/http://lastpage/http://prevpage/http://goback/http://lastpage/http://prevpage/http://goback/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
14/36
Solutions to exercises 14
Full worked solutions
Exercise 1.
RHS = quotient of homogeneous functions of same degree (= 2)
Set y = vx : i.e.d
dx(vx) =
xvx + v2x2
x2
i.e. xdv
dx+ v = v + v2
Separate variables xdv
dx= v2 (subtract v from both sides)
and integrate :
dv
v2=
dx
x
i.e. 1v
= ln x + C
Re-express in terms of x,y : xy
= ln x + C
i.e. y =x
ln x + C
.
Return to Exercise 1Toc Back
http://lastpage/http://prevpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
15/36
Solutions to exercises 15
Exercise 2.
Standard form:dy
dx=
x2 + y2
2xy
i.e. quotient of homogeneous functionsthat have the same degree
Set y = xv:d
dx(xv) =
x2 + x2v2
2x xv
i.e. xdv
dx+
dx
dxv =
x2(1 + v2)
2x2v
i.e. xdv
dx + v =1 + v2
2v
Separate variables
(x, v) and integrate: xdv
dx=
1 + v2
2v v(2v)
(2v)
Toc Back
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
16/36
Solutions to exercises 16
i.e. xdv
dx=
1 v22v
i.e.
2v1 v2 dv =
dxx
Note:d
dv(1 v2) = 2v
i.e.
2v1 v2 dv =
dx
x
i.e. ln(1 v2) = ln x + ln Ci.e. ln[(1 v2)1] = ln(Cx)
i.e.1
1 v2
= Cx
Toc Back
S l i i 17
http://lastpage/http://lastpage/http://lastpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
17/36
Solutions to exercises 17
Re-express in terms of x and y: i.e.1
1 y2x2
= Cx
i.e. x2
x2 y2 = Cx
i.e.x
C= x2 y2 .
Particular solution: x = 1y = 0 gives 1C = 1 0
i.e. C = 1
gives x2
y2 = x .
Return to Exercise 2
Toc Back
S l ti t i 18
http://lastpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
18/36
Solutions to exercises 18
Exercise 3.
Set y = xv:
x
dv
dx + v =
x + xv
x
=x
x(1 + v) = 1 + v
i.e. xdv
dx= 1
Separate variables and integrate:
dv = dx
x
i.e. v = ln x + ln k (ln k = constant)
i.e. v = ln (kx)
Toc Back
S l ti t i 19
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
19/36
Solutions to exercises 19
Re-express in terms of x and y:
y
x
= ln (kx)
i.e. y = x ln (kx) .
Particular solution with y = 1 when x = 1:
1 = ln (k)i.e. k = e1 = e
i.e. y = x ln (ex)
= x[ln e + ln x]
= x[1 + ln x]i.e. y = x + x ln x .
Return to Exercise 3
Toc Back
Solutions to exercises 20
http://lastpage/http://lastpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
20/36
Solutions to exercises 20
Exercise 4.
dydx
= xyx
: Set y = vx: i.e. x dvdx
+ v = 1
v
i.e. x dvdx
= 1 2v i.e. dv12v = dxxi.e. - 12 ln(1 2v) = ln x + ln k
i.e. ln
(1 2v) 12
ln x = ln k
i.e. ln
1
(12v)12 x
= ln k
Toc Back
Solutions to exercises 21
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
21/36
Solutions to exercises 21
i.e. 1 = kx(1 2v) 12
Re-express in x, y: 1 = kx 1 2yx
12
i.e. 1 = kxx2yx
12
(square both sides) 1 = K x2x2yx
, (k2 = K)
i.e. 1 = K x(x
2y)
Particular solution: 1 = K 2 (2 2 12) = K 2 1, i.e. K = 12y(2) = 12 i.e.
x = 2y = 1
2
gives 2 = x2 2xy.
Return to Exercise 4
Toc Back
Solutions to exercises 22
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
22/36
Solutions to exercises 22
Exercise 5.
Set y = xv:x
dv
dx
+ v =x 2xv
x= 1 2v
i.e. xdv
dx= 1 3v
Separate variables and integrate:dv
1 3v =
dx
x
i.e.1
(
3)ln(1 3v) = ln x + ln k (ln k = constant)
i.e. ln (1 3v) = 3 ln x 3 ln ki.e ln (1 3v) + ln x3 = 3 ln ki.e ln [x3(1 3v)] = 3 ln ki.e x3(1
3v) = K (K = constant)
Toc Back
Solutions to exercises 23
http://lastpage/http://lastpage/http://lastpage/http://prevpage/http://prevpage/http://prevpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
23/36
Solutions to exercises 23
Re-express in terms of x and y:
x3 1 3y
x = K
i.e. x3
x 3yx
= K
i.e. x2 (x 3y) = K .
Particular solution with y(1) = 1:
1(1 + 3) = K i.e. K = 4
x2 (x
3y) = 4 .
Return to Exercise 5
Toc Back
Solutions to exercises 24
http://lastpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
24/36
Solutions to exercises 24
Exercise 6.
Already in standard form, with quotient of two first degreehomogeneous functions.
Set y = xv: xdv
dx+ v =
x + vx
x vx
i.e. xdv
dx
=x(1 + v)
x(1 v) v
=1 + v v(1 v)
1 v
i.e. xdv
dx=
1 + v2
1 v
Toc Back
Solutions to exercises 25
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
25/36
Solutions to exercises 25
Separate variables and integrate:
1 v
1 + v2dv =
dx
x
i.e.
dv
1 + v2 1
2
2v
1 + v2=
dx
x
i.e. tan1 v 12
ln(1 + v2) = ln x + A
Re-express in terms of x and y:
tan1y
x
1
2ln
1 +
y2
x2
= ln x + A
i.e. tan1 y
x
=
1
2 lnx2 + y2
x2
+
1
2 ln x2
+ A
=1
2ln
x2 + y2
x2
x2
+ A
Return to Exercise 6
Toc Back
Solutions to exercises 26
http://lastpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
26/36
Exercise 7.
dy
dx =
x2 + y2
2x2Set y = xv:
xdv
dx+ v =
x2 + x2v2
2x2
= 1 + v2
2
i.e. xdv
dx=
1 + v2
2 2v
2
=
1 + v2
2v
2
Toc Back
Solutions to exercises 27
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
27/36
Separate variables and integrate:
dv
1 2v + v2
=1
2
dx
x
i.e.
dv
(1 v)2 =1
2
dx
x
[Note: 1 v is a linear function of v, therefore use standard integraland divide by coefficient of v. In other words,
w = 1 vdwdv
= 1 and dv(1v)2 = 1(1) dww2 .]
i.e. dw
w2 =
1
2 dx
x
i.e.
1w
=
1
2ln x + C
i.e.1
1 v=
1
2
ln x + C
Toc Back
Solutions to exercises 28
http://lastpage/http://lastpage/http://prevpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
28/36
Re-express in terms of x and y:
1
1 yx=
1
2ln x + C
i.e.x
x y =1
2ln x + C
i.e. 2x = (x y)(ln x + C), (C = 2C).
Return to Exercise 7
Toc Back
Solutions to exercises 29
http://lastpage/http://lastpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
29/36
Exercise 8.
dydx
= 2yx2xy . Set y = vx, xdvdx
+ v = 2v12v
x dvdx
= 2v1v(2v)2v =v212v ;
2vv21 dv =
dxx
Partial fractions: 2vv21 =
Av1 +
Bv+1 =
A(v+1)+B(v1)v21
i.e. A + B = 1A B = 2
2A = 1
i.e. A = 12 , B =
32
i.e. 12
1v1 3v+1 dv =
dxx
i.e. 12 ln(v 1) 32 ln(v + 1) = ln x + ln k
Toc Back
Solutions to exercises 30
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
30/36
i.e. ln
(v1)
12
(v+1)32 x
= ln k
i.e.v
1(v+1)3x2 = k
2
Re-express in x, y:
yx
1yx
+ 13
x2= k2
i.e.
yxx
y+xx
3x2
= k2
i.e. y
x = K(y + x)3.
Return to Exercise 8
Toc Back
Solutions to exercises 31
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
31/36
Exercise 9.
RHS is only a function ofv = yx
, so substitute and separate variables.
Set y = xv:
xdv
dx+ v = v + tan v
i.e. xdv
dx
= tan v
Separate variables and integrate:
dv
tanv
= dx
x
{ Note:
cos v
sin vdx
f(v)
f(v)dv = ln[f(v)] + C }
Toc Back
Solutions to exercises 32
http://lastpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
32/36
i.e. ln[sin v] = ln x + ln k (ln k = constant)
i.e. ln
sin vx
= ln k
i.e.sin v
x= k
i.e. sin v = kx
Re-express in terms of x and y: siny
x
= kx.
Return to Exercise 9
Toc Back
Solutions to exercises 33
http://lastpage/http://lastpage/http://prevpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
33/36
Exercise 10.
dy
dx=
y
x + e( yx
)
i.e. RHS is function of v = yx
, only.
Set y = vx:
xdv
dx
+ v = v + ev
i.e. xdv
dx= ev
i.e.
evdv =
dx
x
i.e. ev = ln x + ln k= ln(kx)
i.e. ev = ln(kx)
Toc Back
Solutions to exercises 34
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
34/36
Re-express in terms of x, y:
ey
x = ln(kx)i.e. y
x= ln[ ln(kx)]
i.e. y = x ln[ ln(kx)].
Return to Exercise 10
Toc Back
Solutions to exercises 35
http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
35/36
Exercise 11.
dy
dx =
y
x +
1
x
x2 + y2
=y
x+
1 +
yx
2
[Note RHS is a function of only v =yx , so substitute and separate
the variables]
i.e. Set y = xv:
xdv
dx+ v = v + 1 + v2
i.e. xdv
dx=
1 + v2
Toc Back
Solutions to exercises 36
http://lastpage/http://goback/http://prevpage/http://lastpage/8/8/2019 Ordinary Differential Equations Homogfn
36/36
Separate variables and integrate:
dv
1 + v2=
dx
x
{ Standard integral:
dv1 + v2
= sinh1(v) + C }
i.e. sinh1(v) = ln x + A
Re-express in terms of x and y
sinh1 y
x = ln x + A .Return to Exercise 11
Toc Back
http://lastpage/http://goback/http://prevpage/http://lastpage/