Problem 49 - Advanced Engineering Mathematics

Question

The Laplace transform \(\mathscr{L}\left\\{e^{-r^{2}}\right\\}\) exists, but without finding it solve the initial-value problem $$ y^{\prime \prime}+9 y=3 e^{-t^{2}}, y(0)=0, y^{\prime}(0)=0 $$

Step-by-Step Solution

Answer

The solution is \( y(t) = 3 \int_0^t \sin(3(t-v))e^{-v^2} \, dv \).

Step 1: Transform the Differential Equation

First, apply the Laplace transform to the given differential equation: \( y''(t) + 9y(t) = 3e^{-t^2} \). Recall that the Laplace transform of a second derivative is \( \mathscr{L}\{y''(t)\} = s^2Y(s) - sy(0) - y'(0) \). Since \( y(0) = 0 \) and \( y'(0) = 0 \), this simplifies to \( s^2Y(s) \). Therefore, we have:\[ s^2Y(s) + 9Y(s) = 3\mathscr{L}\{e^{-t^2}\} \].

Step 2: Factor the Equation for Y(s)

Factor the terms with \( Y(s) \) in the equation obtained from the Laplace transform. This results in:\[ Y(s)(s^2 + 9) = 3\mathscr{L}\{e^{-t^2}\} \].

Step 3: Solve for Y(s)

Solve for \( Y(s) \) by dividing both sides by \( s^2 + 9 \):\[ Y(s) = \frac{3}{s^2 + 9} \mathscr{L}\{e^{-t^2}\} \].

Step 4: Identify the Inverse Laplace Transform

Given \( Y(s) = \frac{3}{s^2 + 9} \mathscr{L}\{e^{-t^2}\} \), we need to find the inverse Laplace transform of this expression to solve for \( y(t) \). The expression \( \frac{3}{s^2 + 9} \) represents the Laplace transform of \( \sin(3t) \). Therefore, the inverse Laplace transform of \( Y(s) \) can be expressed as:\[ y(t) = 3 \int_0^t \sin(3(t-v)) f(v) \, dv \]where \( f(t) = e^{-t^2} \).

Step 5: Express the Integral Solution

Replace \( f(v) \) with its corresponding function \( e^{-v^2} \) in the integral to describe the solution:\[ y(t) = 3 \int_0^t \sin(3(t-v))e^{-v^2} \, dv \].

Step 6: Interpret the Solution

This integral expression provides the solution \( y(t) \) for the given initial-value problem. Since \( \mathscr{L}\{e^{-t^2}\} \) was not explicitly calculated, the expression represents a solution in terms of convolution integrals.

Key Concepts

Initial-value problem

An initial-value problem is a type of differential equation accompanied by specific conditions that must be met by the solution. These conditions, known as initial conditions, specify values for the function and its derivatives at a particular point.
In our specific problem, the differential equation is:

• \(y''+9y=3e^{-t^2}\)

and the initial conditions are:

• \(y(0)=0\)
• \(y'(0)=0\)

These initial conditions help us to find a unique solution to the differential equation.
Initial-value problems are common in physics and engineering, as they model how systems evolve from given starting states. Solving them typically involves methods like separation of variables, characteristic equations, or as in this exercise, using Laplace transforms.

Inverse Laplace Transform

The Inverse Laplace Transform is a crucial concept that allows us to move from the Laplace domain back to the time domain. It is essentially the reverse operation of the Laplace Transform. The goal is to retrieve the original function of time which has been transformed.
In the provided exercise, we are given the expression:

• \(Y(s) = \frac{3}{s^2 + 9} \mathscr{L}\{e^{-t^2}\}\)

The task is to find how this expression translates into the time-domain function, \(y(t)\).
The term \(\frac{3}{s^2 + 9}\) is recognized as the Laplace transform of \(3\sin(3t)\). Applying the Inverse Laplace Transform involves calculating:

• \(y(t) = 3 \int_0^t \sin(3(t-v)) e^{-v^2} \ dv\)

Through this process, the inverse yields the function \(y(t)\) in terms of a convolution integral.

Differential Equation

Differential equations are equations that involve an unknown function and its derivatives. They are incredibly important in modeling a wide variety of real-world phenomena such as physical systems in engineering, population dynamics in biology, and financial markets in economics.
In our case, the differential equation:

• \(y'' + 9y=3e^{-t^2}\)

represents a second-order linear differential equation. This type indicates it involves the second derivative of the unknown function \(y\).
The presence of the term \(3e^{-t^2}\) on the right-hand side makes this a non-homogeneous differential equation, as it’s not solely dependent on \(y\) or its derivatives. Solving it typically involves using techniques like the method of undetermined coefficients, variation of parameters, or Laplace transforms.
Converting the differential equation into the Laplace domain can simplify solving it, especially when initial conditions are involved, as it transforms differential equations into algebraic ones. This is why Laplace transforms are such a powerful tool in handling complex initial-value problems.