Problem 10 - Advanced Engineering Mathematics

Question

Write the given system without the use of matrices. $$ \frac{d}{d t}\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{rr} 3 & -7 \\ 1 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)+\left(\begin{array}{l} 4 \\ 8 \end{array}\right) \sin t+\left(\begin{array}{l} t-4 \\ 2 t+1 \end{array}\right) e^{4 t} $$

Step-by-Step Solution

Answer

The individual equations without matrices are: \(\dot{x} = 3x - 7y + 4\sin t + (t-4)e^{4t}\) and \(\dot{y} = x + y + 8\sin t + (2t+1)e^{4t}\)."

Step 1: Express the System

Start by expressing the given system in terms of individual equations for \(x\) and \(y\). The system represents the derivatives \(\frac{d}{dt}\) of \(x\) and \(y\). Here it is written as: \[\frac{d}{dt}\left(\begin{array}{c}x \ y\end{array}\right)=\left(\begin{array}{rr}3 & -7 \ 1 & 1\end{array}\right)\left(\begin{array}{c}x \ y\end{array}\right)+\left(\begin{array}{c}4 \ 8\end{array}\right) \sin t+\left(\begin{array}{c}t-4 \ 2t+1\end{array}\right) e^{4t} \]

Key Concepts

System of Differential Equations

A system of differential equations is a set of equations that relate functions and their derivatives. Imagine you have multiple variables that change over time, like the position of planets or electrical currents in a circuit. These changes over time can be expressed using differential equations. In this specific example, we have a system that describes how two functions, say \(x(t)\) and \(y(t)\), evolve over time.

• Each equation in the system corresponds to one of these functions.
• The equations involve the derivatives of these functions, indicating how they change over time.
• Relationships between functions can be influenced by external factors, such as inputs or forces.

Understanding a system of differential equations helps in analyzing how complex systems behave, predict future states, and determine equilibrium conditions.

Matrix Representation

Matrix representation is a concise way to write systems of linear differential equations. Instead of writing each equation separately, matrices combine everything into a compact form.
For example, in our exercise, the system of equations is written as:\[\frac{d}{dt}\left(\begin{array}{c}x \y\end{array}\right)=\left(\begin{array}{rr}3 & -7 \1 & 1\end{array}\right)\left(\begin{array}{c}x \y\end{array}\right)+\left(\begin{array}{c}4 \8\end{array}\right) \sin t +\left(\begin{array}{c}t-4 \2t+1\end{array}\right) e^{4t}\]

• The matrix on the right of the equality sign is the coefficient matrix. It contains constants that can modify \(x\) and \(y\).
• The vector next to the coefficient matrix represents the changing functions \(x\) and \(y\).
• Additional vectors and functions in the equation provide external influence or driven inputs to the system.

Using matrices simplifies the handling of systems of equations, making calculations more streamlined and visualizing interaction within the system easier.

First-order Differential Equations

First-order differential equations are equations involving the first derivative of a function and no higher derivatives. These types of equations are crucial because they describe rates of change, which are foundational in modeling dynamic systems.
In our exercise, the differential equations for \(x(t)\) and \(y(t)\) involve their first derivatives. Therefore, we classify them as first-order.

• The derivative \(\frac{dx}{dt}\) or \(\frac{dy}{dt}\) gives the rate of change of the function with respect to time \(t\).
• Understanding first-order differential equations helps predict how systems will evolve from an initial state.
• They often model real-world scenarios where the future state depends on the present rate of change.

Grasping first-order differential equations is key to running simulations, making predictions, and understanding fundamental principles in fields ranging from physics to economics.