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Partial Differential Equations

by Adeel
(Pakistan)










































Calculus - Differentials

Find the partial differential equation for the family of planes when the sum of the x, y, and z intercepts is equal to unity.

Comments for Partial Differential Equations

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Jun 28, 2013
Differential Equations
by: Staff


Answer

Part I

Three points define the family planes for this problem:

the x-axis intercept: (a, 0, 0)

the y-axis intercept: (0, b, 0)

the z-axis intercept: (0, 0, c)


Three points define the family of planes





the intercept form of the equation for a family of planes is:

x/a + y/b + z/c = 1


intercept form of the equation for a family of planes







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Jun 28, 2013
Differential Equations
by: Staff


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Part II


For this problem the sum of the three intercepts is equal to unity

a + b + c = 1


the sum of the three intercepts is equal to unity





the partial differential of z with respect to x:

∂z/∂x = p

the partial differential of z with respect to y:

∂z/∂y = q


the partial differential of z with respect to x: 

∂z/∂x = p 

the partial differential of z with respect to y: 

∂z/∂y = q







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Jun 28, 2013
Differential Equations
by: Staff


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Part III


For this problem, the sum of whose x, y, and z intercepts is equal to unity:

a + b +c = 1

c = 1 - a - b


solve for the z-intercept, c






Substitute (1 – a - b) for c in the equation for the family of planes

x/a + y/b + z/c = 1

x/a + y/b + z/(1-a-b) = 1


Substitute  (1 – a - b) for c in the equation for the family of planes








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Jun 28, 2013
Differential Equations
by: Staff


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Part IV


differentiating with respect to x

∂/∂x(x/a + y/b + z/(1-a-b) = 1)

1/a + p/(1-a-b) = 0

which can be rewritten

p/(1-a-b) = -1/a

and

ap = a + b - 1



differentiating with respect to x





differentiating with respect to y

∂/∂y(1/b + q/(1-a-b) = 1)

1/b + q/(1-a-b) = 0


which can be rewritten

q/(1-a-b) = -1/b

and

bq = a + b - 1


differentiating with respect to y






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Jun 28, 2013
Differential Equations
by: Staff


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Part V


dividing the result of differentiating by x with the result of differentiating by y


p/(1-a-b) = -1/a

divided by

q/(1-a-b) = -1/b


p/q = b/a


dividing the result of differentiating by x with the result of differentiating by y


p/q = b/a

since

ap = a + b - 1


then

p = a/a + b/a - 1/a

p = 1 + b/a - 1/a

p = 1 + p/q - 1/a

ap = a + ap/q - 1

1 = a + ap/q - ap

1 = a + ap/q - ap

1 = a(1 + p/q - p)

q = a(q + p - pq)

a = q/(q + p - pq)


using the same reasoning


b = p/(q + p - pq)


solve for the x-intercept, a

 solve for the y-intercept, b








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Jun 28, 2013
Differential Equations
by: Anonymous


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Part VI


we began with this equation:

x/a + y/b + z/(1-a-b) = 1


substituting the values of a and b

(p + q - pq)x/q + (p + q - pq)y/p + (p + q - pq)z/(-pq) = 1

x/q + y/p + z/(-pq) = 1/(p + q - pq)

or

px + qy - z = pq/(p + q - pq)


substitute the values of a and b into the equation for the family of planes





substitute the values of a and b into the equation for the family of planes - continued






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Jun 28, 2013
Differential Equations
by: Staff


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Part VII


substitute the values of a and b into the equation for the family of planes - continued





the final answer is:

px + qy - z = pq/(p + q - pq)


partial differential equation for the family of planes – final answer








Thanks for writing.

Staff
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