A curve C is called a flow line of a vector field F if F is a tangent vector to C at each point along C (See the accompanying figure). (a) Let C be a flow line for F( x , y ) = − y i+ x j, and let ( x , y ) be a point on C for which y ≠ 0. Show that the flow lines satisfy the differential equation d y d x = − x y (b) Solve the differential equation in part (a) by separation of variables, and show that the flow lines are concentric circles centered at the origin.
A curve C is called a flow line of a vector field F if F is a tangent vector to C at each point along C (See the accompanying figure). (a) Let C be a flow line for F( x , y ) = − y i+ x j, and let ( x , y ) be a point on C for which y ≠ 0. Show that the flow lines satisfy the differential equation d y d x = − x y (b) Solve the differential equation in part (a) by separation of variables, and show that the flow lines are concentric circles centered at the origin.
A curve C is called a flow line of a vector field F if F is a tangent vector to C at each point along C (See the accompanying figure).
(a) Let C be a flow line for
F(
x
,
y
)
=
−
y
i+
x
j,
and let
(
x
,
y
)
be a point on C for which
y
≠
0.
Show that the flow lines satisfy the differential equation
d
y
d
x
=
−
x
y
(b) Solve the differential equation in part (a) by separation of variables, and show that the flow lines are concentric circles centered at the origin.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
dy
=x-Y.
Consider the differential equation
dx
a) What is the slope of the direction field at the point (-1.5, -1.5) ?
b) What is the slope of the direction field at the point (-0.5, 1.5) ?
c) What is the slope of the direction field along the x-axis? Enter a number or an expression.
d) What is the slope of the direction field along the y-axis? Enter a number or an expression.
e) Use your answers above to choose the correct direction field for the differential equation.
III/
1117/
1117/
|| | //
-²¹,
111/1.
III/
¹+
111/42-
IQ
1
I
- 1²₁
VAI
NATI
NAII
SI
C
M
Since z = f( a, b, d, c), write the exact differential of the dependent variable.
Show that y1(t) = cos(3t) and y2(t) = sin(3t) are linearly independent for all t.
Thomas' Calculus: Early Transcendentals (14th Edition)
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