Question

Consider the linear homogeneous system $$ \begin{aligned} x^{\prime} &=p_{11}(t) x+p_{12}(t) y \\ y^{\prime} &=p_{21}(t) x+p_{22}(t) y \end{aligned} $$ Show that if \(x=x_{1}(t), y=y_{1}(t)\) and \(x=x_{2}(t), y=y_{2}(t)\) are two solutions of the given system, then \(x=c_{1} x_{1}(t)+c_{2} x_{2}(t), y=c_{1} y_{1}(t)+c_{2} y_{2}(t)\) is also a solution for any constants \(c_{1}\) and \(c_{2} .\) This is the principle of superposition.

Short answer

Answer: The principle of superposition states that the linear combination of any two solutions of a linear homogeneous system of equations is also a solution of that same system. This principle applies to linear homogeneous systems because they satisfy the properties of linearity, i.e., the system remains unchanged by addition of multiple solutions and their scaling by constants. In our example, the linear combination of two given solutions with constant weights \(c_1\) and \(c_2\) formed another solution of the system, thereby demonstrating the principle of superposition.

Step-by-step solution

Write the given system of equations

The system of equations is given as follows: $$ \begin{cases} x' = p_{11}(t) x + p_{12}(t) y \\ y' = p_{21}(t) x + p_{22}(t) y \end{cases} $$

Find the derivatives of the linear combination

We are given the solutions \(x = x_1(t), y = y_1(t)\) and \(x = x_2(t), y = y_2(t)\). If we consider the linear combination of these solutions with constant weights \(c_1\) and \(c_2\), we will get: $$ \begin{cases} x = c_1 x_1(t) + c_2 x_2(t) \\ y = c_1 y_1(t) + c_2 y_2(t) \end{cases} $$ Now, we need to find the derivatives of the linear combination with respect to time \(t\). So, we get: $$ \begin{cases} x' = c_1 x_1'(t) + c_2 x_2'(t) \\ y' = c_1 y_1'(t) + c_2 y_2'(t) \end{cases} $$

Plug the linear combination and its derivatives into the given equations

Now we plug the expressions for \(x\), \(y\), \(x'\), and \(y'\) from the linear combination into the given system of equations: $$ \begin{cases} c_1 x_1'(t) + c_2 x_2'(t) = p_{11}(t)(c_1 x_1(t) + c_2 x_2(t)) + p_{12}(t)(c_1 y_1(t) + c_2 y_2(t)) \\ c_1 y_1'(t) + c_2 y_2'(t) = p_{21}(t)(c_1 x_1(t) + c_2 x_2(t)) + p_{22}(t)(c_1 y_1(t) + c_2 y_2(t)) \end{cases} $$

Show that the principle of superposition holds

We know that \((x_1, y_1)\) and \((x_2, y_2)\) are solutions to the given system of equations. Therefore, the following equations hold: $$ \begin{cases} x_1'(t) = p_{11}(t) x_1(t) + p_{12}(t) y_1(t) \\ y_1'(t) = p_{21}(t) x_1(t) + p_{22}(t) y_1(t) \end{cases} $$ $$ \begin{cases} x_2'(t) = p_{11}(t) x_2(t) + p_{12}(t) y_2(t) \\ y_2'(t) = p_{21}(t) x_2(t) + p_{22}(t) y_2(t) \end{cases} $$ Now we multiply the first set of equations by \(c_1\) and the second set of equations by \(c_2\). Then we add them: $$ \begin{cases} c_1 x_1'(t) + c_2 x_2'(t) = p_{11}(t)(c_1 x_1(t) + c_2 x_2(t)) + p_{12}(t)(c_1 y_1(t) + c_2 y_2(t)) \\ c_1 y_1'(t) + c_2 y_2'(t) = p_{21}(t)(c_1 x_1(t) + c_2 x_2(t)) + p_{22}(t)(c_1 y_1(t) + c_2 y_2(t)) \end{cases} $$ We can see that these are the same equations we got when we plugged the expressions from the linear combination into the given system of equations. Therefore, the principle of superposition holds, and our claim is proven. Thus, \(x = c_1 x_1(t) + c_2 x_2(t), y = c_1 y_1(t) + c_2 y_2(t)\) is also a solution for the given linear homogeneous system.

Key concepts

Linear Homogeneous Systems

A linear homogeneous system comprises a set of linear equations where all the terms are proportional to the unknown function or its derivatives and there is no constant term. In mathematical terms, a system is considered homogeneous if, after setting the dependent variables (e.g., x and y) to zero, the right-hand side of the equations becomes zero. Such systems have a key property where the trivial solution, where all variables are zero, is always a solution.

In the context of differential equations, linear homogeneous systems appear as a set of equations involving derivatives of unknown functions that can be expressed in matrix form. This matrix form makes it easier to apply analytical methods to find solutions for the systems. When solved, these systems can reveal important characteristics of dynamic systems, such as stability and resonance, which are of great interest in fields like engineering and physics.

Differential Equations

Differential equations are mathematical expressions that relate a function to its derivatives. These equations are fundamental in describing physical phenomena where change is continuous—examples include motion, growth, decay, and wave propagation.

Depending on the order of the highest derivative in the equation, differential equations can be classified as first-order, second-order, and so on. The key to solving a differential equation is finding the original function (or functions), best known as the solution of the equation, which satisfies the relationship specified by the equation for as wide a range of the independent variable as possible. Solutions can be complex and may require different techniques to solve, such as separation of variables, integration factors, or Laplace transforms, depending on the form and order of the equation involved.

Solution of Differential Equations

Solving a differential equation means finding a function, or a set of functions, that satisfies the equation for some conditions known as initial or boundary values. The solutions to linear homogeneous differential equations can be especially neat because they allow for the principle of superposition to be applied.

The general solution to a linear differential equation is typically a combination of particular solutions and a homogeneous solution. For instance, in a system of equations, each individual solution reflects a specific pathway relating the variables and their rates of change. When these individual solutions are found, they can be combined into a general solution that encompasses a wide range of initial conditions or system states. This versatility is what makes the study of the solutions to differential equations so significant in science and engineering, where systems are often subject to varying initial conditions.

Superposition Principle in Linear Systems

The superposition principle is a fundamental characteristic of linear systems whereby the sum of any two solutions to a linear equation is also a solution to the equation. This principle applies directly to linear homogeneous systems of differential equations, as demonstrated in the exercise.

The superposition principle enables the construction of general solutions by combining particular solutions in a linear fashion, weighted by constant coefficients. What's truly powerful about this principle is that it simplifies the solution process: it allows for complex systems to be broken down into simpler parts, solved individually, and then reassembled to form the full solution. Ultimately, the principle of superposition is a cornerstone in fields like quantum mechanics, signal processing, and acoustics, as it directly informs the methods used to analyze waves and vibrations in various contexts.