Solving Second-Order ODEs Using Laplace Transform: A Comprehensive Guide and Example

Second-order ordinary differential equations (ODEs) are ubiquitous in many fields of science and engineering. Solving these equations can be challenging, but employing the Laplace transform can provide a systematic and effective approach. This guide will walk you through the process of solving a second-order linear ODE using this powerful technique, including a detailed example.

Step-by-Step Process for Solving Second-Order ODEs with Laplace Transform

The Laplace transform is an invaluable tool in solving second-order ODEs, converting the problem from the time domain into the frequency domain where algebraic techniques can be applied. Here’s a step-by-step guide:

Step 1: Take the Laplace Transform

For a second-order linear ODE of the form:

[a y''(t) b y'(t) c y(t) f(t)]

where (y(t)) is the unknown function, (f(t)) is a given function, and (a), (b), and (c) are constants, apply the Laplace transform to both sides.

The Laplace transform of a function (y(t)) is defined as:

[Y(s) mathcal{L}{y(t)} int_0^{infty} e^{-st} y(t) dt]

Using the properties of the Laplace transform, we have:

[mathcal{L}{y'(t)} sY(s) - y(0)]

[mathcal{L}{y''(t)} s^2Y(s) - sy(0) - y'(0)]

Step 2: Substitute Transforms into the ODE

Substituting the transforms into the ODE gives:

[a s^2Y(s) - a y(0) - a y'(0) b sY(s) - b y(0) c Y(s) F(s)]

where (F(s) mathcal{L}{f(t)}).

Step 3: Rearrange the Equation

Rearranging the equation yields:

[a s^2Y(s) - b sY(s) - cY(s) F(s) - a y(0) - a y'(0) - b y(0)]

[(a s^2 - b s - c)Y(s) F(s) - a y(0) - a y'(0) - b y(0)]

Step 4: Solve for (Y(s))

Isolate (Y(s)):

[Y(s) frac{F(s)}{a s^2 - b s - c} - frac{a y(0) a y'(0)}{a s^2 - b s - c} - frac{b y(0)}{a s^2 - b s - c}]

Step 5: Perform Inverse Laplace Transform

Now take the inverse Laplace transform of (Y(s)) to find (y(t)), either directly or after decomposing (Y(s)) into partial fractions.

Example

Consider the equation:

[y''(t) - 3y'(t) - 2y(t) e^{-t}]

with initial conditions (y(0) 1) and (y'(0) 0).

Step 1: Laplace Transform

Apply the Laplace transform:

[mathcal{L}{y''(t)} - 3mathcal{L}{y'(t)} - 2mathcal{L}{y(t)} mathcal{L}{e^{-t}}]

[s^2Y(s) - s - 0 - 3(sY(s) - 1) - 2Y(s) frac{1}{s 1}]

Step 2: Substitute and Rearrange

Substituting and rearranging gives:

[s^2Y(s) - 3sY(s) - 2Y(s) - s 3 frac{1}{s 1}]

[(s^2 - 3s - 2)Y(s) frac{1}{s 1} s - 3]

Step 3: Solve for (Y(s))

Isolating (Y(s)) yields:

[Y(s) frac{(s 1)(s-3) s(s 1)}{s^2 - 3s - 2}]

[Y(s) frac{s^2 - 3s s - 3 s^2 s}{s^2 - 3s - 2}]

[Y(s) frac{2s^2 - 2s - 3}{s^2 - 3s - 2}]

Step 4: Perform Inverse Laplace Transform

Decompose (Y(s)) into partial fractions and find (y(t)) using known inverse transforms.

Note: The detailed inverse transform is a lengthy process and is best left to symbolic computation software or careful hand calculations.

Why Use Laplace Transforms?

The powerful feature of Laplace transforms is their ability to convert differential equations into algebraic ones. This simplification often reduces a complex problem into a series of manageable algebraic steps. Additionally, extensive tables of Laplace transforms are useful for quickly finding the inverse transforms.

In summary, the Laplace transform provides a systematic approach to solve second-order linear ODEs, leveraging the power of algebra to solve problems in the frequency domain. With a bit of practice, this method can be mastered, making it a valuable tool in the mathematician’s and engineer’s toolkit.