Solving the Constant PDE ∂u/∂x=∂u/∂y

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In summary, the given equation states that the gradient of u in the x direction is equal to the gradient in the y direction for all (x,y). This means that u can be a function of both x and y. The general solution is u(x,y)=f(x+y), where f is a differentiable function. Therefore, u can be dependent on x and y, as shown by the example u=2(x+y). "Ensure" in this context means to guarantee or make certain.
  • #1
AlonsoMcLaren
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∂u/∂x=∂u/∂y, can we ensure that u is a constant not dependent on x and y?
 
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  • #2
It says that the gradient of u in the x direction is the same as in the y direction for all (x,y).
What sort of function has that property?

What do you mean by "ensure"?
 
  • #3
AlonsoMcLaren said:
∂u/∂x=∂u/∂y, can we ensure that u is a constant not dependent on x and y?

The general solution is u(x,y)=f(x+y), where f is a differentiable function.
So u can be dependent on x and y.
For instance u=2(x+y) is a solution.
 

FAQ: Solving the Constant PDE ∂u/∂x=∂u/∂y

What is a PDE and how is it different from an ordinary differential equation (ODE)?

A PDE, or partial differential equation, involves functions of multiple variables and their partial derivatives. This is different from an ODE, which only involves functions of a single variable and their derivatives.

2. What does it mean to solve a PDE?

Solving a PDE means finding a function or set of functions that satisfies the equation for all values of the independent variables. It involves finding a solution that satisfies both the PDE and any specified boundary or initial conditions.

3. Can PDEs be solved analytically or numerically?

PDEs can be solved using both analytical and numerical methods. Analytical solutions involve finding an exact formula for the solution, while numerical solutions involve approximating the solution using algorithms and computational methods.

4. What are the boundary and initial conditions for solving a PDE?

The boundary conditions specify the values of the solution on the boundaries of the domain, while the initial conditions specify the values of the solution at a specific initial time or location. These conditions are necessary for determining a unique solution to the PDE.

5. What are some real-world applications of solving PDEs?

PDEs have many applications in physics, engineering, and other fields. They are used to model phenomena such as heat transfer, fluid dynamics, electricity and magnetism, and many more. PDEs also play a crucial role in mathematical modeling and predicting the behavior of complex systems.

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