Question

Let \(x=x_{1}(t), y=y_{1}(t)\) and \(x=x_{2}(t), y=y_{2}(t)\) be any two solutions of the linear nonhomogeneous system $$ \begin{aligned} x^{\prime} &=p_{11}(t) x+p_{12}(t) y+g_{1}(t) \\ y^{\prime} &=p_{21}(t) x+p_{22}(t) y+g_{2}(t) \end{aligned} $$ Show that \(x=x_{1}(t)-x_{2}(t), y=y_{1}(t)-y_{2}(t)\) is a solution of the corresponding homogeneous system.

Short answer

Answer: The difference between the solutions of the two given nonhomogeneous linear systems is a solution to the corresponding homogeneous system.

Step-by-step solution

Write down the homogeneous system corresponding to the given nonhomogeneous system

The homogeneous system corresponding to the given linear nonhomogeneous system is: $$ \begin{aligned} x^{\prime} &= p_{11}(t)x + p_{12}(t)y, \\ y^{\prime} &= p_{21}(t)x + p_{22}(t)y. \end{aligned} $$

Calculate the derivatives of x and y in terms of x_1, y_1, x_2, y_2, and their derivatives

We are given \(x=x_{1}(t)-x_{2}(t)\) and \(y=y_{1}(t)-y_{2}(t)\). To calculate their derivatives, we differentiate both expressions with respect to t: $$ \begin{aligned} x^{\prime} &= \frac{d}{dt}(x_{1}(t) - x_{2}(t)) = x_{1}^{\prime}(t) - x_{2}^{\prime}(t), \\ y^{\prime} &= \frac{d}{dt}(y_{1}(t) - y_{2}(t)) = y_{1}^{\prime}(t) - y_{2}^{\prime}(t). \end{aligned} $$

Substitute the derivatives and x and y obtained in step 2 into the homogeneous system

Now, let's substitute the expressions for x, y, x', and y' that we found in step 2 into the homogeneous system: $$ \begin{aligned} x_{1}^{\prime}(t) - x_{2}^{\prime}(t) &= p_{11}(t)(x_{1}(t)-x_{2}(t)) + p_{12}(t)(y_{1}(t)-y_{2}(t)), \\ y_{1}^{\prime}(t) - y_{2}^{\prime}(t) &= p_{21}(t)(x_{1}(t)-x_{2}(t)) + p_{22}(t)(y_{1}(t)-y_{2}(t)). \end{aligned} $$

Simplify the expressions and prove that they satisfy the homogeneous system

To simplify the expressions, we can substitute the derivatives of x_1, y_1, x_2, and y_2 from the given nonhomogeneous system: $$ \begin{aligned} (p_{11}(t)x_{1}(t) + p_{12}(t)y_{1}(t) + g_{1}(t)) - (p_{11}(t)x_{2}(t) + p_{12}(t)y_{2}(t) + g_{1}(t)) &= p_{11}(t)(x_{1}(t)-x_{2}(t)) + p_{12}(t)(y_{1}(t)-y_{2}(t)), \\ (p_{21}(t)x_{1}(t) + p_{22}(t)y_{1}(t) + g_{2}(t)) - (p_{21}(t)x_{2}(t) + p_{22}(t)y_{2}(t) + g_{2}(t)) &= p_{21}(t)(x_{1}(-t)-x_{2}(t)) + p_{22}(t)(y_{1}(t)-y_{2}(t)). \end{aligned} $$ Notice that the terms with g_1(t) and g_2(t) cancel out on each equation, thus simplifying to: $$ \begin{aligned} p_{11}(t)x_{1}(t) + p_{12}(t)y_{1}(t) - p_{11}(t)x_{2}(t) - p_{12}(t)y_{2}(t) &= p_{11}(t)(x_{1}(t)-x_{2}(t)) + p_{12}(t)(y_{1}(t)-y_{2}(t)), \\ p_{21}(t)x_{1}(t) + p_{22}(t)y_{1}(t) - p_{21}(t)x_{2}(t) - p_{22}(t)y_{2}(t) &= p_{21}(t)(x_{1}(t)-x_{2}(t)) + p_{22}(t)(y_{1}(t)-y_{2}(t)). \end{aligned} $$ These equations are true, so we can conclude that \(x=x_{1}(t)-x_{2}(t), y=y_{1}(t)-y_{2}(t)\) is a solution of the corresponding homogeneous system.

Key concepts

Homogeneous Systems

A homogeneous system of differential equations is a set where each equation is homogeneous in terms of the unknown variables. This means that each equation in the system equates to zero when there are no external forces or inputs, often denoted by zero functions. The general form involves only the dependent variables and their derivatives:

• Equation form: In the case of two variables, the standard form is:\[\begin{aligned}x' &= a(t)x + b(t)y,y' &= c(t)x + d(t)y \end{aligned}\]
• Characteristics: A homogeneous system lacks any additional "forcing" or non-zero terms.
• Solution behavior: Solutions to homogeneous systems typically exhibit behavior such as oscillations or exponential growth/decay that are solely dictated by the coefficients of the system.

Understanding that solutions to homogeneous systems are foundational in analyzing more complicated nonhomogeneous systems is crucial. The principle of superposition often applies, meaning linear combinations of solutions are also solutions.

Solution Verification

To verify a solution to a system of equations, one must substitute the proposed solution back into the original equations. For the exercise, this involves the solutions \(x=x_{1}(t)-x_{2}(t)\) and \(y=y_{1}(t)-y_{2}(t)\).

• Substitution: By plugging in these expressions, derived from solutions to the nonhomogeneous system, into the homogeneous counterpart, you assess the correctness of the solution.
• Simplification: Simplifying the equations helps confirm that no additional terms or errors persist.
• Consistency: A consistent solution will lead to true statements after simplification.

The cancellation of nonhomogeneous elements \(g_1(t)\) and \(g_2(t)\) demonstrates that the hypothesis holds, thus verifying your solution accurately represents the behavior prescribed by the homogeneous system.

Differential Equations

Differential equations form the backbone of modeling dynamic systems where changes occur over time. They consist of derivatives which express rates of change.

• Types: These can be - Ordinary Differential Equations (ODEs), involving a single variable. - Partial Differential Equations (PDEs), involving multiple variables.
• Order: The order of the differential equation is determined by the highest derivative present.
• Linear vs. Nonlinear: Linear differential equations have solutions that form a vector space, while nonlinear ones do not adhere to this constraint.

For our exercise, we focused on a linear nonhomogeneous system of ODEs involving two variables. The core challenge was to demonstrate how the subtraction of solutions transforms it into a homogeneous system, emphasizing the linearity and its impact on the system's structure.

System of Equations

A system of equations is a collection of multiple equations that are solved together, each sharing common variables. They can be classified based on their homogeneity and linearity.

• System Elements: Each system comprises equations that share dependent variables, often with differing relationships or constraints between them.
• Solving Strategies: - Substitution: Solving one equation for a variable and substituting that into another. - Elimination: Adding or subtracting equations to eliminate a variable. - Matrix Methods: Utilizing matrices to represent and solve complex systems.
• Linear Systems: These have solutions that exist in a plane or hyperplane in the space of independent variables.

Understanding systems of equations aids in grasping how different equations interact and how their solutions are interconnected. Particularly, how transforming a nonhomogeneous system into a homogeneous form can simplify the solution process.