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Zinaida [17]
3 years ago
7

b) If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations

dx/dt = x and dy/dt = −y. Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1). , where x 0
Mathematics
1 answer:
sineoko [7]3 years ago
4 0

Answer:

The equation of the the flow line that passes through the point (x, y) = (−1, −1) is

In y + In x = 0 or in another form, xy = 1.

Step-by-step explanation:

The pathline equation for a vector field is given by F(x,y) = xî - yj

The velocity vector field for the streamline of the flow is given by

V(x, y) = (dx/dt)î + (dy/dt)j

From the question, it is given that

(dx/dt) = x

(dy/dt) = -y

Hence, the velocity vector field for the streamline of the flow in question is

V(x, y) = xî - yj

which coincides with the pathline vector field of the flow.

The only time the pathline and streamline vector field coincide and have the same equation is when the flow is a steady state flow.

That is, the properties of the fluid flowing isn't changing with time!

Hence, this flow is a steady state flow!

We're told to solve the differential equation.

(dx/dt) = x

(dy/dt) = -y

but

(dy/dx) = (dy/dt) × (dt/dx)

(dy/dx) = -y/x

(dy/y) = -(dx/x)

∫(dy/y) = -∫ (dx/x)

In y = - In x + c

where c is the constant of integration

In y + In x = c

In (xy) = c

Inserting the values of (x, y) given in the question,

In (-1 × -1) = c

In 1 = c

0 = c

c = 0

In y + In x = 0

In (yx) = 0

xy = e⁰ = 1

xy = 1

So, the equation of the the flow line that passes through the point (x, y) = (−1, −1) is

In y + In x = 0 or in another form, xy = 1

Hope this Helps!!!

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Mariana [72]
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Information Given:
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ON = 7x - 9
LM = 6x + 4
MN = x - 7
OL = 2y - 7

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Since it is a parallelogram:
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ON = LM and
MN = OL

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ON = LM:
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7x - 9 = 6x + 4

-----------------------------------------------
Subtract 6x from both sides:
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x - 9 = 4

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Add 9 to both sides:
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x = 13

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MN = OL:
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x - 7 = 2y - 7

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Sub x = 13:
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13 - 7 = 2y - 7

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Simplify:
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6 = 2y - 7

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Add 7 on both sides:
-----------------------------------------------
13 = 2y

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Divide by 2:
-----------------------------------------------
y = 13/2

-----------------------------------------------
Answer: x = 13, y = 13/2 (Answer D)
-----------------------------------------------


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