Separation of Variables

Separation of variables (also known as the Fourier approach) is a method for solving ordinary and partial differential equations that uses algebra to rewrite an equation such that each of two variables appears on a different side of the equation. To solve a first order or first degree differential equation, there are three approaches. Only the solution of differential equations with variables separated will be discussed.

Ordinary Differential Equation (ODE)

An ordinary differential equation contains the function and its derivatives.  It just has one independent variable and one or more derivatives of that variable.

Ordinary differential equations have an order that is determined by the highest derivative in the equation. The n^th order ODE has the following general form:

F(x, y, y’,…., yⁿ ) = 0

Let us consider and ordinary differential equation

(dy/dx) = f(x)g(y)

The functions f and g each depend on only one variable in this unusual example. With this criterion met, we can now “separate” x-related phrases from y-containing expressions. We can achieve this by rewriting the equation with the functions f and g on opposing sides. Divide by g(y) first:

(1/g(y)) . (dy/dx) = f(x)

Upon multiplying both sides with “dx”, we get,

(dy/g(y)) = f(x) . dx

All terms linked to y are now on the left-hand side of the equation, whereas all terms related to x are on the right-hand side. As a result, both sides of the equation can now be integrated.

∫(dy/g(y)) =∫f(x) . dx

Depending on which functions are in the equation, the differential equation can now be solved analytically or numerically.

Partial Differential Equations (PDE)

The separation of variables technique may solve partial differential equations as well as regular differential equations. Consider the heat equation, which is one of the most well-known partial differential equations. Joseph Fourier, a mathematician who worked around the turn of the century, was the first to formulate the heat equation. The concept of differential equations was still in its infancy at the time. As a result, Fourier’s hypothesis took a long time to catch on. Fourier proposed that the rate at which a solid’s temperature changes is proportional to the pace at which heat diffuses through it.

(∂u/∂t) = α(∂2u/∂x2)

The temperature change over time would be zero if the heat was distributed equally over the solid. A significant influx of heat from a solid location, on the other hand, would swiftly raise the temperature of neighboring areas.

Let’s split down the heat equation into ordinary differential equations, which are easier to understand, using the separation of variables technique. We’ll have to utilize a process known as “ansatz” to do this. Ansatz, a Greek term that means “guess,” can assist us in finding a solution. It just so happens that the proper ansatz for the heat equation is a separable function.

u(x,T) = X(x) . T(t)

We get the partial differential equation by substituting this ansatz into it.

u(x,T) = X(x) . T(t)

(∂(xT)/∂T) = α(∂2(xT)/∂x2)

XT’ = αX’’T

We may now distinguish the phrases that only contain x from those that only contain t once again.

T’/T = αX”/X

Following that, I’ll assert that either side of this equation must be constant. While this may appear odd at first, consider that the two sides have no variables in common. Changing the value of t, in other words, only changes the left-hand side of the equation. Maintaining the value of x, on the other hand, will only influence the right-hand side of the equation.

As a result, we can write that both equations are equivalent to k.

k = T’/T = αX”/X

This step is crucial because the original partial differential equation may now be decomposed into two ordinary differential equations, each with a constant, k.

T’= kT

X” = (k/α)X

The benefit of dividing our equation in this manner is that we can now use all of the other methods for solving ordinary differential equations. There are many more of them than there are ways to solve a partial differential equation.

Finally, the separation of variables methodology is a straightforward analytical technique that may be used to solve both ordinary and partial differential equations.

Conclusion

Differential equations include variables as well as the derivative of the dependent variable (y) with respect to the independent variable (x) in addition to the dependent and independent variables. Differential equations stated in terms of (x,y) with x-terms and y-terms on opposite sides of the equation (including delta terms). As a result, each variable may be readily integrated to generate a differential equation solution.