Variación de parámetros

What is the method of variation of parameters?

When it comes to solving non-homogeneous differential equations, the method of variation of parameters is an invaluable tool. Unlike the method of indeterminate coefficients, which is only applicable to certain specific functions, the method of variation of parameters allows us to deal with any non-homogeneous equation. This method makes it easier to find the particular solution needed to complete the general solution of the equation.

How is the non-homogeneous equation represented?

Let us begin by defining how a non-homogeneous differential equation is represented in its general form. It is structured as follows:

[ a(x) \cdot y'' + b(x) \cdot y' + c(x) \cdot y = d(x) ].

For simplicity, we divide the entire equation by ( a(x) ), allowing us to work with simple derivatives:

[ y'' + p(x) \cdot y' + q(x) \cdot y = f(x) ]

Where:

• ( p(x) = \frac{b(x)}{a(x)} )
• ( q(x) = \frac{c(x)}{a(x)} )
• ( f(x) = \frac{d(x)}{a(x)} )

This initial step leads us to work comfortably with the concepts of the method.

What is the general solution to the homogeneous equation?

The related homogeneous equation is obtained by equating the right-hand side of the non-homogeneous equation to zero:

[ y'' + p(x) \cdot y' + q(x) \cdot y = 0 ].

The general solution for this homogeneous equation includes two linearly independent solutions:

[ y_h = c_1 \cdot y_1 + c_2 \cdot y_2 ]

Here, ( y_1 ) and ( y_2 ) are independent solutions and ( c_1 ) and ( c_2 ) are constants. The interesting thing is that, in the parameter variation method, these constants will be replaced by functions, to find the particular solution.

How is the particular solution found?

The particular solution is assumed to have the form:

[ y_p = u_1(x) \cdot y_1 + u_2(x) \cdot y_2 ]

Where ( u_1(x) ) and ( u_2(x) ) are functions to be determined. Equations derived from the original system using the Wronskian are used.

What is the Wronskian?

The Wronskian of two functions is a crucial algebraic concept for the method. It is the determinant of a 2x2 matrix that relates the functions to their derivatives:

[ W(y_1, y_2) = \begin{vmatrix} y_1 & y_2 \ y_1' & y_2' \end{vmatrix} = y_1 \cdot y_2' - y_1' \cdot y_2 ]

How are ( u_1 ) and ( u_2 ) determined?

The function ( u_1(x) ) is found by the formula:

[ u_1(x) = -frac{y_2 \dot f(x)}{W(y_1, y_2)} , dx ]

And the function ( u_2(x) ):

[ u_2(x) = \int \frac{y_1 \dot f(x)}{W(y_1, y_2)} , dx ]

With these equations, we find ( u_1(x) ) and ( u_2(x) ) to substitute into the particular solution.

How is the complete solution composed?

Finally, the complete and general solution of the non-homogeneous differential equation will be the sum of the solution of its homogeneous part and its particular solution:

[ y = y_h + y_p = c_1 \cdot y_1 + c_2 \cdot y_2 + u_1(x) \cdot y_1 + u_2(x) \cdot y_2 ]

This process, although detailed, becomes a powerful tool that allows us to tackle complex differential equations that would otherwise be impossible to solve with basic methods. We hope that the instructions and steps described will be of great use in tackling future mathematical problems.