17.4.1. General Form of Simultaneous Linear Differential Equations

A typical system of two linear differential equations:

$\frac{dx}{dt} = a_{11} x + a_{12} y + f_1(t)$
$\frac{dy}{dt} = a_{21} x + a_{22} y + f_2(t)$

Where:

• x(t) and y(t) are functions of time,
• a_ij are constants,
• f_1(t), f_2(t) are known functions.

In this section, we define the general form of simultaneous linear differential equations. These equations represent systems with multiple interrelated variables; in this case, they involve two functions, x(t) and y(t). The equations depict how the rate of change of x and y depends linearly on each other as well as on some known external functions (f1(t) and f2(t)). The coefficients a_ij are constants that determine how strongly one variable influences the other.

Consider a pair of interconnected water tanks where the flow of water from one tank to another is controlled by constants that reflect the size of the pipes. Here, the levels of water (x and y) in these tanks change over time based on the flow rates (f1(t) and f2(t)) which can vary, just like the functions in our equations.

1. Take Laplace Transform of both equations assuming initial conditions x(0), y(0).

2. Use the property: $L\{\frac{dx}{dt}\} = sX(s) - x(0)$, $L\{\frac{dy}{dt}\} = sY (s) - y(0)$

3. Form a system of algebraic equations in terms of X(s) and Y(s).

4. Solve the algebraic equations using substitution or elimination.

5. Apply Inverse Laplace Transform to get the solution x(t) and y(t).

The steps outline how to solve simultaneous linear differential equations using the Laplace Transform method. First, we transform the differential equations into algebraic equations by applying the Laplace Transform, which simplifies handling derivatives. Next, we substitute the Laplace variable representations (X(s) and Y(s)) into the transformed equations to create a system of algebraic equations. We then solve for the unknowns (X(s) and Y(s)) using algebraic techniques, such as substitution or elimination. Finally, we revert back to the time domain using the Inverse Laplace Transform to find x(t) and y(t), the original functions of time.

Imagine you're trying to figure out the trajectory of two cars (x and y) moving in relation to each other based on their current speed and direction (the differential equations). By translating their paths into a 'map' (the Laplace Transform), we can easily find out where they will be at any future time, instead of tracking every moment of their movement directly.

Problem: Solve the system using Laplace Transform:

$\frac{dx}{dt} = 3x + 4y,$ $\frac{dy}{dt} = -4x + 3y,$

$x(0)=1, y(0)=0$

1. Take Laplace Transforms ...

Final Answer: $x(t) = e^{3t} \cos(4t), \, y(t) = -e^{3t} \sin(4t)$

In this example, we demonstrate how to apply the steps outlined previously to solve a specific system of linear differential equations. The equations are transformed using the Laplace Transform, and we arrive at a system of equations in the s-domain. We then manipulate these equations, applying algebraic techniques to isolate variables and solve for X(s) and Y(s). Finally, we utilize the Inverse Laplace Transform to return to the time domain and find x(t) and y(t), thus completing the solution process.

Think of this example as a detective story where you're given initial clues (the initial conditions) about two suspects (x and y). By transforming the mysteries (the differential equations) into a more manageable format, you piece together the evidence (the algebraic equations) leading to the final outcome (the functions x(t) and y(t)) that solve the case.

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