Question

How do you come up with an ordinary differential equation from a partial differential equation? Illustrate using the one-dimensional wave equation.

Answer 1

To come up with an ordinary differential equation from a partial differential equation, one must eliminate the partial derivatives. This can be done by assuming the solution has a specific form and then substituting it into the partial differential equation. For example, in the one-dimensional wave equation, assuming the solution has the form y(x,t) = f(x)g(t), we can rewrite the equation and separate it into two ordinary differential equations.

Step-by-step explanation:

To come up with an ordinary differential equation from a partial differential equation, one must eliminate the partial derivatives. This can be done by assuming the solution has a specific form and then substituting it into the partial differential equation. For example, in the one-dimensional wave equation:

∂²y/∂t² = c²(∂²y/∂x²)

Assuming the solution has the form y(x,t) = f(x)g(t), we can rewrite the equation as:

g''(t)f(x) = c²f''(x)g(t)

Dividing both sides by f(x)g(t) gives:

g''(t)/g(t) = c²f''(x)/f(x)

Since the left side depends only on time and the right side depends only on position, the equation can be separated into two ordinary differential equations:

g''(t)/g(t) = k²

f''(x)/f(x) = k²/c²

Where k is a constant. These are ordinary differential equations that can be solved separately to find the solution to the original partial differential equation.

Answer 2

To come up with an ordinary differential equation (ODE) from a partial differential equation (PDE), you can consider the PDE in one dimension and assume that the dependent variable is a function of only one variable. By doing this, you can then eliminate the partial derivatives and convert the PDE into an ODE.

Step-by-step explanation:

To come up with an ordinary differential equation (ODE) from a partial differential equation (PDE), you can consider the PDE in one dimension and assume that the dependent variable is a function of only one variable. By doing this, you can then eliminate the partial derivatives and convert the PDE into an ODE.

For example, let's consider the one-dimensional wave equation:

∂²u/∂t² = c²∂²u/∂x²

If we assume that the dependent variable u(x,t) is a function of only the variable x, we can write u(x,t) = f(x). Substituting this into the wave equation, we get:

∂²f/∂t² = c²∂²f/∂x²

This is now an ordinary differential equation.