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. $$ Y''(y) + \frac{kY(y)}{x+y} = 0 $$

Key concepts

Partial Differential Equations

Partial Differential Equations, commonly referred to as PDEs, are equations that involve partial derivatives of a function with more than one independent variable. This means that the function we're dealing with is dependent on multiple variables and must satisfy the given equation across these variables. A partial differential equation is essentially a way to describe how a multivariable function changes with changes in its variables. To better comprehend PDEs, consider how temperature might change on a metal plate. In this case, the temperature could depend on the plate's position (two variables, x and y) and time. The heat equation, a type of PDE, helps us describe this temperature distribution over time and space. PDEs can model a wide range of phenomena, from heat and wave propagation to quantum mechanics and fluid dynamics. Understanding PDEs involves identifying the equation's order, which tells us the highest derivative present. It is also important to explore the equation's linearity, as linear PDEs are generally simpler to solve compared to nonlinear ones. In practice, finding solutions to PDEs often requires specialized mathematical techniques, such as separation of variables, which simplifies solving complex equations by breaking them down into separate, solvable parts.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) involve functions of a single independent variable and their derivatives. This simplification from PDEs to ODEs means that the function is only dependent on one variable, making these equations generally easier to solve.For instance, if a PDE that describes two spatial dimensions is broken down using separation of variables, it can lead to a pair of ODEs. Each ODE then describes the change of the solution with respect to a single variable, such as time or space.When using the separation of variables method, we often reformulate PDEs into one or more ODEs by setting parts of the equation equal to a constant. With an ODE like \[X''(x) - kX(x) = 0\]we can focus solely on one variable. Solving ODEs typically involves standard techniques, like integrating (or antidifferentiating) functions or using characteristic equations for linear cases. In summary, ODEs provide a simpler path to finding solutions once a complex PDE is reduced through methods like separation of variables.

Separable Solutions

The concept of separable solutions is a fundamental technique used to solve PDEs by reducing them to ODEs. A solution is considered *separable* if a function of multiple variables can be expressed as the product of single-variable functions.For example, in the PDE given by \[u(x, y) = X(x)Y(y)\] we assume that the function can be written as a combination of two independent functions, one depending only on x and the other on y.The process of separation involves rewriting the PDE to separate the functions of x and y. Once managed, you can break the PDE into more manageable, independent ODEs. It's crucial in the separation process to equate the parts of the equation, enabling each to depend on just one variable, typically by using constants.If successfully separated, as shown with: \[\frac{X''(x)}{X(x)} = k = -(x+y)\frac{Y''(y)}{Y(y)}\] we can solve these ODEs separately. This way, separable solutions allow for complex, multidimensional problems to be tackled one dimension at a time, simplifying the approach significantly.