The direction field for dx = 5 (a) Verify that the straight lines y = 3x are solution curves, provided x#0. = (b) Sketch the solution curve with initial condition y(0) = -1. (c) Sketch the solution curve with initial condition y(3) = 1. (d) What can be said about the behavior of the above solutions as x→+co? How about x→-∞o? 25x 9y (a) The restriction y#0 is needed because dy dx (Simplify your answer.) Now let y = is shown to the right. - Substituting the expression for y into the differential equation yields = dy 25x dx 9 The result from the previous step simplifies to (Simplify your answer.) (Simplify your answer.) 25x 9y is not defined when y = 0. Consider the straight lines one at a time. First let y= 5 3x x. In this case, the standard rules of differentiation yield dx 5 verifying that the straight line y = 3x is a solution curve. - .... 25x dy = Substituting the expression for y into the differential equation yields dx 9
The direction field for dx = 5 (a) Verify that the straight lines y = 3x are solution curves, provided x#0. = (b) Sketch the solution curve with initial condition y(0) = -1. (c) Sketch the solution curve with initial condition y(3) = 1. (d) What can be said about the behavior of the above solutions as x→+co? How about x→-∞o? 25x 9y (a) The restriction y#0 is needed because dy dx (Simplify your answer.) Now let y = is shown to the right. - Substituting the expression for y into the differential equation yields = dy 25x dx 9 The result from the previous step simplifies to (Simplify your answer.) (Simplify your answer.) 25x 9y is not defined when y = 0. Consider the straight lines one at a time. First let y= 5 3x x. In this case, the standard rules of differentiation yield dx 5 verifying that the straight line y = 3x is a solution curve. - .... 25x dy = Substituting the expression for y into the differential equation yields dx 9
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The direction field for
dy
dx
25x
gy
is shown to the right.
(a) Verify that the straight lines y = ±x are solution curves, provided x#0.
3
(b) Sketch the solution curve with initial condition y(0) = -1.
(c) Sketch the solution curve with initial condition y(3) = 1.
(d) What can be said about the behavior of the above solutions as x→ + co? How about x→-co?
(a) The restriction y#0 is needed because
dy
dx
(Simplify your answer.)
25x
9y
is not defined when y = 0. Consider the straight lines one at a time. First let y=
dy
Substituting the expression for y into the differential equation yields
dx 9
-
5
The result from the previous step simplifies to verifying that the straight line y=x is a solution curve.
(Simplify your answer.)
5
Now let y = -x. In this case, the standard rules of differentiation yield
3
(Simplify your answer.)
25x
dy
Substituting the expression for y into the differential equation yields dx
dy
II
25x
O
90
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