Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > Let f(Z) = u(x, y) + i(v(x, y)) be an analyti...
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Let f(Z) = u(x, y) + i(v(x, y)) be an analytical function. If u = 5x + 2xy then v is equal to _________, where c is a constant.
- a)x2 + y2 - 5x + c
- b)x2 - y2 - 5xy + c
- c)x2 + y2 + 5xy + c
- d)y2 - x2 + 5y + c
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
Let f(Z) = u(x, y) + i(v(x, y)) be an analytical function. If u = 5x +...
Concept:
Let f(Z) = u(x, y) + i(v(x, y)) be an analytical function.
If the real part u(x, y) of analytic function f(z) is given then to find imaginary part v(x, y) of f(z) we can use the following procedure:
1). Find ux and uy
2). Consider dv =
Let f(Z) = u(x, y) + i(v(x, y)) be an analytical function.
If the real part u(x, y) of analytic function f(z) is given then to find imaginary part v(x, y) of f(z) we can use the following procedure:
1). Find ux and uy
2). Consider dv =
3). dv = vx dx + vy dy = -uy dx + ux dy.
4). v = ∫(−uy)dx + ∫(ux)dy + c
Calculation:
Given:
u = 5x + 2xy
ux = 5 + 2 × y, uy = 2 × x.
dv = vx dx + vy dy = -uy dx + ux dy = (-2 × x) dx + (5 + 2 × y) dy.
Given:
u = 5x + 2xy
ux = 5 + 2 × y, uy = 2 × x.
dv = vx dx + vy dy = -uy dx + ux dy = (-2 × x) dx + (5 + 2 × y) dy.
Now by integrating we get:
v = ∫(−uy) dx + ∫(ux)dy
v = ∫(−2x) dx + ∫(5+2y)dy + c
y2 - x2 + 5y + c
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Let f(Z) = u(x, y) + i(v(x, y)) be an analytical function. If u = 5x +...
Understanding the Analytical Function
Given the function f(Z) = u(x, y) + i(v(x, y)), where u = 5x + 2xy, we need to find the corresponding v using the Cauchy-Riemann equations.
Cauchy-Riemann Equations
The Cauchy-Riemann equations state:
1. ∂u/∂x = ∂v/∂y
2. ∂u/∂y = -∂v/∂x
Calculate the Partial Derivatives of u
- ∂u/∂x = ∂(5x + 2xy)/∂x = 5 + 2y
- ∂u/∂y = ∂(5x + 2xy)/∂y = 2x
Apply the Cauchy-Riemann Equations
Using the first Cauchy-Riemann equation:
- ∂v/∂y = 5 + 2y
Using the second Cauchy-Riemann equation:
- -∂v/∂x = 2x → ∂v/∂x = -2x
Integrate to Find v
Now we integrate to find v:
1. Integrate ∂v/∂y with respect to y:
- v = (5y + y^2) + g(x), where g(x) is a function of x.
2. Differentiate the result with respect to x:
- ∂v/∂x = g'(x) → set it equal to -2x.
- g'(x) = -2x → Integrating gives g(x) = -x^2 + C, where C is a constant.
Final Expression for v
Thus, substituting g(x) back into the equation for v:
v = 5y + y^2 - x^2 + C
To match the format of the options provided and rearranging, we have:
v = -x^2 + y^2 + 5y + C
Hence, the correct answer is option 'D': y^2 - x^2 + 5y + c.
Given the function f(Z) = u(x, y) + i(v(x, y)), where u = 5x + 2xy, we need to find the corresponding v using the Cauchy-Riemann equations.
Cauchy-Riemann Equations
The Cauchy-Riemann equations state:
1. ∂u/∂x = ∂v/∂y
2. ∂u/∂y = -∂v/∂x
Calculate the Partial Derivatives of u
- ∂u/∂x = ∂(5x + 2xy)/∂x = 5 + 2y
- ∂u/∂y = ∂(5x + 2xy)/∂y = 2x
Apply the Cauchy-Riemann Equations
Using the first Cauchy-Riemann equation:
- ∂v/∂y = 5 + 2y
Using the second Cauchy-Riemann equation:
- -∂v/∂x = 2x → ∂v/∂x = -2x
Integrate to Find v
Now we integrate to find v:
1. Integrate ∂v/∂y with respect to y:
- v = (5y + y^2) + g(x), where g(x) is a function of x.
2. Differentiate the result with respect to x:
- ∂v/∂x = g'(x) → set it equal to -2x.
- g'(x) = -2x → Integrating gives g(x) = -x^2 + C, where C is a constant.
Final Expression for v
Thus, substituting g(x) back into the equation for v:
v = 5y + y^2 - x^2 + C
To match the format of the options provided and rearranging, we have:
v = -x^2 + y^2 + 5y + C
Hence, the correct answer is option 'D': y^2 - x^2 + 5y + c.
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