Variational Methods

Boundary value problems are a complex form of first order equation problems that have the solution in multiple (most commonly two) points on the function curve. The points on the function that the solution provides are the boundary values of the function. 

When we solve a boundary problem, we initially get a standard first order linear differential equation and eventually specify a function at various points on the curve. Although obtaining particular values of the various points is not a vital step in every case, the values play a vital role in the solution of the function in some way. We typically use various formulas from mathematical sciences to solve the first order equation and find a function value that satisfies the conditions for boundary points.

First Order Differential Equation

Order of a differential equation is the highest power of the primary variable in the term of the equation. So, a first order linear differential equation is an equation that has 1 as its highest power. Solving first order differential equation is easier when compared to the methods of solving equations of higher order. 

Differential equations are written in the form F(t, y, y’) = 0, where y is a variable function of t, and y’ is a derivative of y. We use the first order equation to solve various problems in calculus. First order differential equations are the basic building blocks of calculus knowledge and the foundation of modern mathematical sciences.

Solving First Order Differential Equations

There are two main methods for solving first order differential equations – 

Using the integrating factor:

The standard form for a linear differential equation is [dy/dx + P(x) y = Q(x)]

So, the integrating factor of the given equation will be [u(x) = exp (∫a(x)dx)]

Multiplying u(x) with the content on the left side, we get the product [y(x)u(x)].

The general solution of the differential equation is expressed as follows:

The value of y = ex + C (x3)

[C is a constant (arbitrary constant of integration).]

Using a variation of a Constant:

Like in the integrating factor method, we can find the general solution of a differential equation using a constant variation. The first step in this method is to find the general solution of a homogeneous equation.

Let’s assume our equation is [y’ + a(x)y = 0]

We know that a general solution of every homogeneous equation will consist of C, where C is a constant (arbitrary constant of integration). We replace the C with C(x) for the variation of the constant method, where C(x) is an unknown function.

Next, we will substitute C(x) solution into the non-homogeneous first order differential equation. This substitution will let us find the value of C(x).

Whether we use the integrating factor method or the variation of the constant method, both lead to the same values or set of values for solving first order differential equation. However, it is better to use the integrating factor in the initial stage of learning as it also leads to a better understanding of integration and arbitrary constants.

Definition of Variational Method

The variational method is widely used in wave theory. Mathematical sciences use this method to minimize the value of a wave through variations. The variational principle is the foundation theory for the variational method. We use the variational method to calculate the wavelength (approximate) in quantum mechanics due to its efficiency in finding the ground state (the eigenstate with the minimum energy) and the number of excited states in an orbital.

Variational Methods for Boundary Value Problems

As mentioned above, the second method of solving the first order differential equation is the variation of constants. We can also solve the first order linear differential equations using the variational principle. As we know, we can write every boundary value problem in the form of a differential equation having a set of solutions that abide by the boundary conditions. Hence, we can solve a boundary problem by writing the problem in terms of a differential equation and solving the equation using the variational method.

Conclusion

Boundary value problems can be easily solved using the formulas we use for solving first order differential equations. The concept of boundary values and variational methods have enriched the way we study modern calculus. The solution to the problems is relatively easy, considering that the formulas used to solve them are very common in mathematical sciences.

Differential equations are the essential elements of advanced calculus, and there are numerous ways to solve them. However, the discovery of boundary value problems and variational methods have added more prominence to the concept. We can now easily find the value of boundary points in a function using the formulas to solve differential equations.