Separation of Variables Method

Introduction 

The separation of variables method is one of the ways to solve several types of differential equations. Leibnitz discovered this method of solving equations. He also developed the process of solving the homogeneous differential equations of the first order. This method can be used on ordinary or partial differential equations. Other analytic solution techniques like direct integration are best used on simple ODEs. In contrast, the separation of variables method is easier to apply to more difficult Ordinary differential equations and even partial differential equations. It can also help solve equations that are linear and homogeneous. 

What are Differential Equations? 

Before moving forward, it is important to briefly look back at essential concepts like differential equations, partial differential equations, and ordinary differential equations.

Whether a differential equation is known as a partial differential equation (PDE) or an ordinary differential equation (ODE) depends on whether they contain partial derivatives.

The highest order derivative comes first in the order of a differential equation. 

Differential equations are statements that contain one or multiple derivatives terms. This refers to terms that represent the rates of change of fluctuating quantities. Differential equations are very commonly used in science and engineering.

The basis of separation of variables method

The separation variables method is based on the understanding that if g(y) and f(x) are independent variables y and x. Additionally if,

 f (x) = g (y)

Then there is must be a constant which is equal to both f(x) and g(y)

•  δ/δx f (x) = δ/δx g(y) = 0 ⇒ f’ (x) = 0 ⇒ f(x) is a constant
•  δ/δt g(y) = δ/δt f(x) = 0 ⇒ g’(y) = 0 ⇒ g(x) is a constant

After separating variables, the solution should be of the separated form – 

 u (x, y) = X(x) Y(y)

We then substitute the separated form of the equation back into the original equation and move all x terms to one side and all y terms to the other.  

After separating, the two sides should be constant, requiring the results to be an ordinary differential equation. 

The theory of separation of variables method 

The separation of variables method attempts to solve the equation by rearranging the terms in a pre-described standard manner and then separating the variables. The pre-described standard form being – 

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

The result is then found by dividing the standard form by g(y) and then integrating the terms concerning x. 

It is important to keep non-homogenous, non-linear, and boundaryless equations in mind. Sometimes, even equations that are homogenous, linear, and have boundaries won’t be solvable through this method. We call these types of equations non-separable differential equations.

Example 1.

Find the general solution of equation dy/dx = x + 2/ 4 – y 

Solution: Given, dy/dx = x + 2/ 4 – y

After separating we get,

(4 – y) dy = dx (x + 2)

integrating both sides

∫ (4− y) dy = ∫( 2 + x) dx

= 4y – y2/ 2 = x2/2 + x + C (c is the arbitrary constant)

= 8y – 2y2 = 2×2 + 2x + 2c

= – 2y2 – 2×2 + 8y – 2x + 2c = 0

= – y2 – x2 + 4y – x + c = 0

Example 2. 

Separate variables and integrate: dy/dx = xy/x+1

The numerator and denominator here have the same degree in x: reduce the degree of numerator using long division. 

  x/ x+1 = x+1−1/ x+1 = x+1/ x+1 − 1/ x+1 = 1 − 1/ x+1 

  ∫(1 −1 /x + 1) dx = ∫ dy/ y 

  x − ln(x + 1) = ln y + ln k (ln k = constant of integration)

  x = ln(x + 1) + ln y + ln k = ln[ky(x + 1)]

ex = ky(x + 1) .

The Pros and Cons of the Separation of Variables Method

The advantages of the separation method are: It is simple, direct, easy to understand, and easy to solve. However, this is not always the case in complex problems like the boundary value. The separation of variables method becomes convoluted and hard to solve in such instances.

The separation of variables method is simple and effective. But it has a few limitations. Listed below are some of them. The equations that can be solved with the separation method are limited.

• The equation must be linear, as the solution is found as a sum.
• The coefficients cannot be too complex for the equation to be separable. Separating variables is a must. A coefficient like a sin(XY) in the equation is too complex and not separable.
• The boundaries must remain at constant values. For example, in the heat conduction of a bar, each end must be at a specific, permanent location, such as x = 1 and x = l. The bar cannot expand or move in any way, as that would make the values of the ends dependent on time.

Conclusion

The separation of variables is a method of solving a partial differential equation and ordinary differential equations. This method is easy and quick. It is based on the fact that f(x) = g(y). Not all differential equations can be solved through this method. Even some linear and homogeneous equations can’t be solved through this method. It is also important to note that, in general, a solution arrived at from the separation method isn’t the only solution to the form. The separation of variables method study material is a pathway to understanding the separation method, but it also helps you deepen your understanding of the differential equations themselves,