Question

Use Laplace Transforms to solve the initial value problem where

[y +3y=f(x),y(0)=0]f(x)={ 2/4 0≤x and x<2 2≤x
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Answer 1

The provided question involves solving an initial value problem using Laplace Transforms for a piecewise function defining the non-homogeneous term.

Step-by-step explanation:

The question requires the use of Laplace Transforms to solve the initial value problem given by y'' + 3y = f(x), where y(0)=0 and the piecewise function f(x) is defined as 2 for 0 ≤ x < 2 and 0 for x ≥ 2. To solve this, start by taking the Laplace transform of both sides of the differential equation. Denote L{y''} as s^2 Y(s) - sy(0) - y'(0) and L{y} as Y(s), where Y(s) is the Laplace transform of y(t).

Next, apply the transform to the right side considering f(x) as a piecewise function. The Laplace transform of the step function is utilized for the portion where f(x) is equal to 2. Write out the resultant algebraic equation, solve for Y(s), and then apply the inverse Laplace transform to find y(t). Remember to consider the initial condition y(0) = 0 when solving for constants. Finally, apply the step function in the inverse Laplace to account for the change in f(x) at x=2.