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What is the complete solution for the equation x (y - z) p + y (z - x) q = z (x - y)?
- a)ϕ (x + y + z, xyz) = 0
- b)ϕ (x + 2y + z, xz) = 0
- c)ϕ (2x + y + z, xyz) = 0
- d)ϕ (x + y + z, xy) = 0
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
What is the complete solution for the equation x (y - z) p + y (z - x)...
Concept:
Linear Partial Differential Equation of First Order: A linear partial differential equation of the first order, commonly known as Lagrange's Linear equation, is of the form Pp + Qq = R where P, Q and R are functions of x, y, z. This equation is called a quasi-linear equation. When P, Q and R are independent of z it is known as linear equation.
Thus, to solve the equation of the form Pp + Qq = R
Linear Partial Differential Equation of First Order: A linear partial differential equation of the first order, commonly known as Lagrange's Linear equation, is of the form Pp + Qq = R where P, Q and R are functions of x, y, z. This equation is called a quasi-linear equation. When P, Q and R are independent of z it is known as linear equation.
Thus, to solve the equation of the form Pp + Qq = R
- form the subsidiary equations as dx/P = dy/Q = dz/R
- solve these simultaneous equations by any the method giving u = a and v = b as its solutions.
- write the complete solution as φ (u, v) = 0 or u = f (v).
Calculations:
Given partial differential equation, x(y − z)p + y(z − x)q = z(x − y)
Hence the subsidiary equation is,
So, using the property,
, it can be written as
Hence the subsidiary equation is,
So, using the property,
, it can be written as ∴ dx + dy + dz = 0 and by integrating we get, x + y + z = a
For the second function, using multipliers as 1/x, 1/y, 1/z and using the property,
we get,
we get, and by integrating we get, lnx + lny + lnz = lnb ⇒ xyz = b
Hence, the solution is φ(u, v) = 0 ⇒ φ(x + y + z, xyz) = 0
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Community Answer
What is the complete solution for the equation x (y - z) p + y (z - x)...
To solve the equation x(y - z)p y(z - x)q = z(x - y), we need to simplify the equation and find the values of x, y, and z that satisfy the equation.
Given equation: x(y - z)p y(z - x)q = z(x - y)
Simplifying the equation:
xy^2p - xz^2p + y^2zq - yx^2q = zx - zy
xy^2p - xz^2p + y^2zq - yx^2q - zx + zy = 0
Now let's break down the options and see which one satisfies the equation:
a) (x y z, xyz) = 0
This option suggests that the values of x, y, and z multiplied together should be equal to zero. However, this condition does not ensure that the given equation is satisfied. Therefore, option 'A' is not the correct solution.
b) (x 2y z, xz) = 0
This option suggests that the values of x and z multiplied together should be equal to zero. Again, this condition does not guarantee that the equation is satisfied. Therefore, option 'B' is not the correct solution.
c) (2x y z, xyz) = 0
This option suggests that the values of 2x, y, and z multiplied together should be equal to zero. Similar to the previous options, this condition does not lead to the solution of the equation. Therefore, option 'C' is not the correct solution.
d) (x y z, xy) = 0
This option suggests that the values of x, y, and z multiplied by xy should be equal to zero. However, this condition does not satisfy the equation. Therefore, option 'D' is not the correct solution.
Therefore, the correct solution for the equation x(y - z)p y(z - x)q = z(x - y) is option 'A', which states that (x y z, xyz) = 0.
Given equation: x(y - z)p y(z - x)q = z(x - y)
Simplifying the equation:
xy^2p - xz^2p + y^2zq - yx^2q = zx - zy
xy^2p - xz^2p + y^2zq - yx^2q - zx + zy = 0
Now let's break down the options and see which one satisfies the equation:
a) (x y z, xyz) = 0
This option suggests that the values of x, y, and z multiplied together should be equal to zero. However, this condition does not ensure that the given equation is satisfied. Therefore, option 'A' is not the correct solution.
b) (x 2y z, xz) = 0
This option suggests that the values of x and z multiplied together should be equal to zero. Again, this condition does not guarantee that the equation is satisfied. Therefore, option 'B' is not the correct solution.
c) (2x y z, xyz) = 0
This option suggests that the values of 2x, y, and z multiplied together should be equal to zero. Similar to the previous options, this condition does not lead to the solution of the equation. Therefore, option 'C' is not the correct solution.
d) (x y z, xy) = 0
This option suggests that the values of x, y, and z multiplied by xy should be equal to zero. However, this condition does not satisfy the equation. Therefore, option 'D' is not the correct solution.
Therefore, the correct solution for the equation x(y - z)p y(z - x)q = z(x - y) is option 'A', which states that (x y z, xyz) = 0.
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What is the complete solution for the equation x (y - z) p + y (z - x) q = z (x - y)?a) (x + y + z, xyz) = 0b) (x + 2y + z, xz) = 0c) (2x + y + z, xyz) = 0d) (x + y + z, xy) = 0Correct answer is option 'A'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about What is the complete solution for the equation x (y - z) p + y (z - x) q = z (x - y)?a) (x + y + z, xyz) = 0b) (x + 2y + z, xz) = 0c) (2x + y + z, xyz) = 0d) (x + y + z, xy) = 0Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the complete solution for the equation x (y - z) p + y (z - x) q = z (x - y)?a) (x + y + z, xyz) = 0b) (x + 2y + z, xz) = 0c) (2x + y + z, xyz) = 0d) (x + y + z, xy) = 0Correct answer is option 'A'. Can you explain this answer?.
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