# Partial Differential Equations

(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

 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) the intercept form of the equation for a family of planes is: x/a + y/b + z/c = 1 --------------------------------------------------------------

 Jun 28, 2013 Differential Equations by: Staff -------------------------------------------------------------- Part II For this problem the sum of the three intercepts is equal to unity a + b + c = 1 the partial differential of z with respect to x: ∂z/∂x = p the partial differential of z with respect to y: ∂z/∂y = q --------------------------------------------------------------

 Jun 28, 2013 Differential Equations by: Staff -------------------------------------------------------------- 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 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 --------------------------------------------------------------

 Jun 28, 2013 Differential Equations by: Staff -------------------------------------------------------------- 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 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 --------------------------------------------------------------

 Jun 28, 2013 Differential Equations by: Staff -------------------------------------------------------------- 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 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) --------------------------------------------------------------

 Jun 28, 2013 Differential Equations by: Anonymous -------------------------------------------------------------- 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) --------------------------------------------------------------

 Jun 28, 2013 Differential Equations by: Staff -------------------------------------------------------------- Part VII the final answer is: px + qy - z = pq/(p + q - pq) Thanks for writing. Staff www.solving-math-problems.com