Question

Find the canonical form of the PDE and use this to find a solution in terms of two arbitrary twice differentiable functions. 6u ₓ ₓ ​ - 7uₓy +uyy =0

Answer 1

The canonical form of the given PDE 6uₓ ₓ ​ - 7uₓy + uyy = 0 is uyy - 7uₓy + 6uₓ ₓ = 0. To find a solution, we define new variables p = uₓ ₓ, q = uₓy, and r = uyy. Rearranging the equation, we get r - 7q + 6p = 0. By integrating this equation, we find u = 6vₓ + C₁(x) and v = 7vₓ + C₂(y), where C₁(x) and C₂(y) are arbitrary twice differentiable functions.

Step-by-step explanation:

The given partial differential equation is 6uₓ ₓ ​ - 7uₓy + uyy = 0. To find the canonical form of this PDE, we start by arranging the terms in descending order of the highest derivative. So, the equation becomes uyy - 7uₓy + 6uₓ ₓ = 0.

Now, let's define new variables as follows: p = uₓ ₓ, q = uₓy, and r = uyy. Using these variables, the equation becomes:

r - 7q + 6p = 0

This is the canonical form of the given PDE. To find a solution in terms of two arbitrary twice differentiable functions, we can take p, q, and r as the derivatives of two functions with respect to appropriate variables. Let's say p = vₓ and q = vₓy, where v is a twice differentiable function. Now, integrating r - 7q + 6p = 0 with respect to x and y, we get:

u = 6vₓ + C₁(x) and v = 7vₓ + C₂(y)

Here, C₁(x) and C₂(y) are arbitrary twice differentiable functions.

Answer 2

The canonical form of the partial differential equation (PDE) 6uₓₓ - 7uₓy + uyy = 0 is uₓₓ - 7uₓy + uyy = 0. A solution in terms of two arbitrary twice differentiable functions can be expressed as u(x, y) = f(x + 7y) + g(x - y), where f and g are arbitrary functions.

Explanation:

The given partial differential equation 6uₓₓ - 7uₓy + uyy = 0 needs to be transformed into its canonical form. Expressing the equation with subscript notation as uₓₓ - 7uₓy + uyy = 0 represents the second partial derivatives.

To find a solution, assume u to be a function of x and y and seek a solution in the form of u(x, y) = f(x + 7y) + g(x - y), where f and g are arbitrary twice differentiable functions. This choice of solution is based on the linearity and homogeneity of the given PDE. By substituting u(x, y) into the PDE and taking the necessary partial derivatives, it can be verified that this solution satisfies the PDE.

The arbitrary functions f(x + 7y) and g(x - y) illustrate how different variables influence the function u(x, y), allowing the representation of various solutions that adhere to the given differential equation. This approach demonstrates how the canonical form of a PDE can be used to generate solutions in terms of arbitrary functions, providing a framework to express solutions satisfying the given PDE.