Find the Domain y*((du)/(dx))-x*(du)/(dy)=0

Problem

yd(u)/d(x)−xd(u)/d(y)=0

Solution

1. Identify the type of equation. This is a first-order linear homogeneous partial differential equation (PDE) of the form A(x,y)∂(u)/∂(x)+B(x,y)∂(u)/∂(y)=0

2. Determine the coefficients of the partial derivatives. In this equation, the coefficient of d(u)/d(x) is y and the coefficient of d(u)/d(y) is −x

3. Analyze the domain of the coefficients. The functions A(x,y)=y and B(x,y)=−x are polynomial functions.

4. Conclude the domain. Since polynomial functions are defined and continuous for all real values of x and y there are no restrictions such as division by zero or square roots of negative numbers.

5. State the domain in terms of the Cartesian plane. The equation is well-defined for all points (x,y) in the set of real numbers squared.

Final Answer

Domain=ℝ^2

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