Problem 21 - Advanced Engineering Mathematics

Question

Verify that the vector \(\mathbf{X}_{p}\) is a particular solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=x+4 y+2 t-7 \\ &\frac{d y}{d t}=3 x+2 y-4 t-18 ; \quad \mathbf{X}_{p}=\left(\begin{array}{r} 2 \\ -1 \end{array}\right) t+\left(\begin{array}{l} 5 \\ 1 \end{array}\right) \end{aligned} $$

Step-by-Step Solution

Answer

\( \mathbf{X}_{p} \) satisfies both equations, thus it's a particular solution.

Step 1: Understand the function we need to verify

We need to check if \( \mathbf{X}_{p} = \begin{pmatrix} 2t + 5 \ -t + 1 \end{pmatrix} \) satisfies the given system of differential equations. Our task involves verifying that if we substitute \( \mathbf{X}_{p} \) into the system, it should make both differential equations true.

Step 2: Differentiate \( \mathbf{X}_{p} \) with respect to \( t \)

Calculate the derivative of \( \mathbf{X}_{p} \) with respect to \( t \):\[ \frac{d}{dt} \begin{pmatrix} 2t + 5 \ -t + 1 \end{pmatrix} = \begin{pmatrix} \frac{d}{dt}(2t + 5) \ \frac{d}{dt}(-t + 1) \end{pmatrix} = \begin{pmatrix} 2 \ -1 \end{pmatrix}. \]

Step 3: Substitute \( \mathbf{X}_{p} \) into the first differential equation

Substitute \( x = 2t + 5 \) and \( y = -t + 1 \) into the first equation:\[ \frac{d x}{d t} = 2 = (2t + 5) + 4(-t + 1) + 2t - 7. \]Simplify the right-hand side:\[ (2t + 5) - 4t + 4 + 2t - 7 = 2t + 5 - 4t + 4 + 2t - 7 = 2. \]The equation holds true.

Step 4: Substitute \( \mathbf{X}_{p} \) into the second differential equation

Substitute \( x = 2t + 5 \) and \( y = -t + 1 \) into the second equation:\[ \frac{d y}{d t} = -1 = 3(2t + 5) + 2(-t + 1) - 4t - 18. \]Simplify the right-hand side:\[ 6t + 15 - 2t - 2 - 4t - 18 = -1. \]The equation holds true.

Step 5: Conclusion

Both differential equations are satisfied after substituting \( \mathbf{X}_{p} \). This verifies that \( \mathbf{X}_{p} \) is indeed a particular solution of the system.

Key Concepts

System of Differential Equations

Systems of differential equations are groups of equations that involve unknown functions and their derivatives. These systems model a wide variety of complex, real-world phenomena where multiple variables change over time and are interdependent. For example, such systems can describe:

• Population dynamics in ecological models.
• Electrical circuits with interacting currents and voltages.
• Mechanical systems with several moving parts like in robotics.

Each equation in the system can show how one variable changes relative to others, often written in terms of derivatives with respect to time. Solving these systems requires finding functions for all variables that simultaneously satisfy all equations.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and differential operations applied to them. In the context of differential equations, vectors can represent solutions where each component of the vector corresponds to one of the unknown functions in the system. Here, vector \(\mathbf{X}_{p} = \begin{pmatrix} 2t + 5 \ -t + 1 \end{pmatrix}\) consists of two functions: one for each variable in the differential system.

Differentiating a vector is similar to differentiating each component individually. Thus, vectors provide a compact way to work with multiple equations and variables at once, making it easier to apply and check solutions against the system of equations. This approach is especially helpful in higher-dimensional analyses where vector operations simplify otherwise complex manipulations.

Differential Equations Verification

Verification of differential equations involves ensuring that a given solution satisfies all the equations in a system. The process typically involves:

• Calculating the derivatives of the assumed solution.
• Substituting these derivatives and the functions back into the original differential equations.
• Checking if the resulting expressions hold true for all values within the considered interval.

In our example, by differentiating\(\mathbf{X}_{p}\),we obtained\(\begin{pmatrix} 2 \ -1 \end{pmatrix}\),which we then substituted back into the original equations. As both simplified equations were valid, this confirmed\(\mathbf{X}_{p}\)as a particular solution. Verification is crucial in ensuring the accuracy of solutions derived analytically or computationally.

Particular Solutions

A particular solution to a differential equation is a specific solution that satisfies not only the differential equation but also any additional initial or boundary conditions. In the context of systems, a particular solution like\(\mathbf{X}_{p}\)provides explicit functions for each variable that obey the given equations.

It's important to distinguish this from a general solution, which represents a family of solutions determined by integrating the system and incorporating arbitrary constants. The particular solution emerges from applying initial conditions or specific constraints, providing a precise description of the system's behavior under specified conditions. Thus, in contexts like engineering or physics, particular solutions offer practical, real-world predictions.