Laplace Transformation# Dirac Delta and Systems of ODEs

These are two of the most powerful applications of the Laplace transform. The Dirac delta function lets you model an instantaneous shock, and the method for systems of ODEs lets you solve complex, coupled dynamics using simple algebra.

1. The Dirac Delta Function (The “Impulse”)

The Dirac delta function is not a true function but an “idealized function” or distribution. Its job is to represent an instantaneous impulse — a “hammer strike” or a sudden shock that happens at one exact moment and is immediately over.

Think of it this way:

2. The “Sifting” Property (Its Most Important Trait)

The defining feature of the delta function is what happens when you integrate it with another function, f(t). The delta function “sifts” through the entire function and plucks out the single value of f(t) at the exact moment the impulse occurs.

3. Laplace Transform of the Dirac Delta

Now, finding its Laplace transform is incredibly easy using this property.

By the very definition of the Laplace transform:

4. Systems of Ordinary Differential Equations

The Laplace transform method you’ve learned extends perfectly to systems of coupled ODEs. The process is the same, but instead of one algebra equation, you get a system of algebra equations.

The “Magic”: A system of coupled calculus equations for x(t) and y(t) becomes a system of coupled algebraic equations for X(s) and Y(s).

5. Example: Solving a Coupled System

Let’s solve this two-equation system:

With initial conditions: x(0)=1 and y(0)=0.

Step 1: Transform Every Equation

Step 2: Solve the System (Pure Algebra)

Step 3: Inverse Transform to Find the Solution

Happy learning!!!