Question

Determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations. $$ u_{x x}+(x+y) u_{y y}=0 $$

Short answer

Based on the given partial differential equation (PDE), we determined that the method of separation of variables can be applied to replace the PDE with a pair of ordinary differential equations (ODEs). The ODEs for the given PDE are: 1. $$ X''(x) - kX(x) = 0 $$ 2. $$ Y''(y) + \frac{kY(y)}{x+y} = 0 $$

Step-by-step solution

Assume a separable solution and substitute back in the PDE

Assume that the solution u(x,y) can be separated into two individual functions, one of x and one of y, i.e., u(x,y) = X(x)Y(y). Now, substitute this back into the given PDE: $$ X''(x)Y(y) + (x+y)X(x)Y''(y) = 0 $$ Here, X''(x) and Y''(y) represent the second derivatives of X(x) and Y(y) with respect to x and y, respectively.

Divide the equation by X(x)Y(y)

Divide the equation by X(x)Y(y) to separate the variables: $$ \frac{X''(x)}{X(x)} + (x+y)\frac{Y''(y)}{Y(y)} = 0 $$

Reorganize the equation

Now, rewrite the equation as follows to make it more obvious that the left side is a function of x alone and the right side is a function of y alone: $$ \frac{X''(x)}{X(x)} = -(x+y)\frac{Y''(y)}{Y(y)} $$

Set each side equal to a constant

Since the left side of the equation is a function of x only and the right side is a function of y only, it's necessary for both sides to be equal to a constant for the equation to hold true. Let's call this constant k: $$ \frac{X''(x)}{X(x)} = k = -(x+y)\frac{Y''(y)}{Y(y)} $$ Now, we have two ordinary differential equations: 1. $$ \frac{X''(x)}{X(x)} = k $$ 2. $$ -(x+y)\frac{Y''(y)}{Y(y)} = k $$ Thus, the method of separation of variables can be applied to the given partial differential equation, and the pair of ordinary differential equations are: 1. $$ X''(x) - kX(x) = 0 $$ 2. $$ Y''(y) + \frac{kY(y)}{x+y} = 0 $$