91Ó°ÊÓ

A differential equation is a mathematical equation that relates a function to its derivatives. It tells us how a quantity changes over time or space. In our exercise, the differential equation is given by \( y^{\prime \prime}(x) + y(x) = 4 e^{x} \). Here, \( y(x) \) is the function we want to find. The terms

• \( y^{\prime \prime}(x) \): The second derivative of \( y \), indicating how the rate of change of the rate of change of \( y \) is evolving.
• \( y(x) \): The function itself.
• \( 4e^x \): A function of \( x \) that isn't related to \( y \). It's an external force acting on the system.

Differential equations can be tricky because they often involve finding a balance between multiple terms. Solving them requires finding a function \( y(x) \) that satisfies the equation. This process can involve several steps, like using specific methods such as the Laplace transform.

An initial value problem is a type of differential equation that also includes conditions specifying the value of the function and possibly its derivatives at a particular point. In this exercise, we are given:

• \( y(0) = 0 \)
• \( y^{\prime}(0) = 0 \)

These conditions are essential because they ensure a unique solution to our differential equation. Initial conditions help us pin down exact values at a starting point (usually \( x = 0 \)). Given these, the problem becomes figuring out the function \( y(x) \) that not only satisfies the differential equation but also starts at these specific values. These conditions often help in eliminating arbitrary constants that naturally arise when solving differential equations.

The inverse Laplace transform is a powerful technique used to find the original function from its Laplace-transformed counterpart. This is crucial when solving differential equations in the Laplace domain. In the step-by-step solution of our exercise, \( Y(s) = \frac{4}{(s-1)(s^2+1)} \) was determined as the Laplace transform of the unknown function \( y(t) \).

After transforming into the \( s \)-domain (a domain used for simplifying equations), we retrieve \( y(t) \) by performing the inverse Laplace transform. The process requires translating functions from a complex \( s \)-variable back to a real-world \( t \)-variable.

To do this, we use transformation tables and properties that match common \( s \)-domain forms to their corresponding time-domain functions. Finally, this lets us express \( y(t) \) in terms of familiar functions, such as exponentials and trigonometric expressions.

Partial Fraction Decomposition is a method used to break down complex fractions into simpler ones that are easier to work with, especially when applying inverse transformations like the Laplace transform.

In our solution, \( \frac{4}{(s-1)(s^2+1)} \) needs simplifying to perform the inverse Laplace transform. This involves rewriting it as:

• \( \frac{A}{s-1} + \frac{Bs+C}{s^2+1} \)

The idea is to determine constants \( A, B, \) and \( C \), and equate coefficients after combining terms. By doing these steps, you convert the complex fraction to simpler components related to forms for which inverse Laplace transforms can be easily found. Each part corresponds to a standard inverse transform, which collectively bring us back to the original function in the time domain.