rin 29. dy dx 4.x + 3y 2x + y
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
29

Transcribed Image Text:N 25. Consider the equation mortion besesnqxo nadw vinsup
dy y - 4x misoliamsdam a
onsmedie
(29)
dx
x - y
2200010
a. Show that equation (29) can be rewritten as
=
dy
dx
or
=
(y/x) - 4
1 − (y/x) '
thus equation (29) is homogeneous.
V
y/x, or
b. Introduce a new dependent variable y so that y =
y = xv(x). Express dy/dx in terms of x, v, and dv/dx.
c. Replace y and dy/dx in equation (30) by the expressions fordib.
from part b that involve v and dv/dx. Show that the resulting
differential equation is
dv
v+x- =
dx
dv
-
x dx
=
v-4
1 - v'
²-4
1-v
siquiae
The method outlined in Problem 25 can be used for any
to homogeneous equation. That is, the substitution y = xv(x) transforms
(30)
0) a homogeneous equation into a separable equation. The latter equation
can be solved by direct integration, and then replacing v by y/x
gives the solution to the original equation. In each of Problems 26
through 31:
(31)
Observe that equation (31) is separable.
020 1,0
d. Solve equation (31), obtaining v implicitly in terms of x.
e. Find the solution of equation (29) by replacing v by y/x in
the solution in part d.
f. Draw a direction field and some integral curves for
equation (29). Recall that the right-hand side of equation (29)
actually depends only on the ratio y/x. This means that integral
curves have the same slope at all points on any given straight line
26.
27.
28.
29.
30.
10007
through the origin, although the slope changes from one line to
another. Therefore, the direction field and the integral curves are
symmetric with respect to the origin. Is this symmetry property
evident from your plot?
31.
a. Show that the given equation is homogeneous.
Solve the differential equation.
Gc. Draw a direction field and some integral curves. Are they
symmetric with respect to the origin?
x² + xy + y²
x2
x² + 3y²
a ada als als ala
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
2.3 Modeling with First-Order Differential Equations 39
2xy
4y - 3x
2x - y
4x + 3y
2x + y
x² - 3y²
2xy
x3y²-x²
2xy
=
||
=
ballquesto ba
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