17.6. Final Answer

Application of Laplace Transforms to Simultaneous Linear Differential Equations

In engineering and applied mathematics, systems modeled by simultaneous linear differential equations often arise when dealing with interconnected physical systems like electrical circuits, mechanical systems, or control applications. Solving these equations directly through classical approaches can be complicated and time-consuming. This is where Laplace transforms come into play, offering a systematic method for solving equations algebraically in the s-domain, which drastically simplifies the process.

Objectives

• Convert a system of simultaneous linear differential equations into algebraic equations using Laplace transforms.
• Solve for the unknown functions via algebraic manipulation.
• Retrieve the time-domain solutions utilizing inverse Laplace transforms.

Theoretical Framework

Simultaneous linear differential equations can typically be represented as:

$\frac{dx}{dt} = a_{11} x + a_{12} y + f_1(t)$
$\frac{dy}{dt} = a_{21} x + a_{22} y + f_2(t)$
where $x(t)$ and $y(t)$ are time functions, and $a_{ij}$ are constants, while $f_1(t)$ and $f_2(t)$ are known functions.

Steps to Solve Using Laplace Transforms

1. Take Laplace transforms of both equations assuming initial conditions $x(0)$ and $y(0)$.
2. Apply differentiation properties: $L\{\frac{dx}{dt}\} = sX(s) - x(0)$ and $L\{\frac{dy}{dt}\} = sY(s) - y(0)$.
3. Formulate the system of algebraic equations in terms of $X(s)$ and $Y(s)$.
4. Utilize substitution or elimination to solve the algebraic equations.
5. Finally, apply inverse Laplace transforms to obtain $x(t)$ and $y(t)$.

Solved Example

Given the system:

$\frac{dx}{dt} = 3x + 4y$
$\frac{dy}{dt} = -4x + 3y$
with initial conditions $x(0) = 1, y(0) = 0$.

This section details a systematic procedure to derive $x(t)$ and $y(t)$, illustrating the step-by-step transformations involved.

Summary

• Simultaneous linear differential equations are prevalent in systems with multiple interacting variables.
• Laplace transforms transform these systems into solvable algebraic equations.
• Initial conditions are integrated smoothly during the transformation process.
• Inverse Laplace transforms retrieve time domain solutions, crucial for applications in control systems and engineering dynamics.