The direction field for dy dx y is shown to the right. (a) Verify that the straight lines y = ±x are solution curves, provided x 0. (b) Sketch the solution curve with initial condition y(0) = 2. (c) Sketch the solution curve with initial condition y(4)=1. (d) What can be said about the behavior of the above solutions as x→+00? How about x--00?
The direction field for dy dx y is shown to the right. (a) Verify that the straight lines y = ±x are solution curves, provided x 0. (b) Sketch the solution curve with initial condition y(0) = 2. (c) Sketch the solution curve with initial condition y(4)=1. (d) What can be said about the behavior of the above solutions as x→+00? How about x--00?
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Transcribed Image Text:dy x
The direction field for =
dx
y
is shown to the right.
(a) Verify that the straight lines y = tx are solution curves, provided x #0.
(b) Sketch the solution curve with initial condition y(0) = 2.
(c) Sketch the solution curve with initial condition y(4) = 1.
(d) What can be said about the behavior of the above solutions as x→ +00?
How about x→→-00?
A
(a) The restriction y#0 is needed because is not defined when y=0. Consider the straight lines one at a time. First
dy
dx
let y = x. In this case, the standard rules of differentiation yield
(Simplify your answer.)
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