Question

In Exercises 18 and 19, use dsolve to obtain the solution of each of the indicated second order differential equations. Use the simple command to find the simplest form of that solution. Use ezplot to sketch the solution on the indicated time interval. $$ \begin{aligned} &y^{\prime \prime}+4 y=3 \cos 2.1 t \\ &y(0)=0, y^{\prime}(0)=0,[0,64 \pi] \end{aligned} $$

Short answer

Solve the differential equation, apply initial conditions, find the simplest form, and plot over the interval \([0, 64\pi]\).

Step-by-step solution

Define the Differential Equation

The given differential equation is a second-order linear differential equation.Given:\[y^{\prime \prime} + 4y = 3 \cos(2.1t)\]Boundary conditions:\[y(0) = 0, \quad y^{\prime}(0) = 0\]

Solve the Differential Equation

Use the function `dsolve` to solve the differential equation with the initial conditions. The general solution will have the form:\[y(t) = C_1 \cos(2t) + C_2 \sin(2t) + y_p(t)\]where \(y_p(t)\) is a particular solution to the non-homogeneous part \(3\cos(2.1t)\).

Apply Initial Conditions

Using the initial conditions \(y(0) = 0\) and \(y^{\prime}(0) = 0\), determine the constants \(C_1\) and \(C_2\). This yields a specific solution for the problem.

Substitute and Simplify the Solution

Insert the values of \(C_1\) and \(C_2\) back into the general solution, and then simplify to obtain the simplest form of the solution, denoted as \(y(t)\). In this case, use the command `simple` in software to simplify the expression.

Sketch the Solution Graph

Use the function `ezplot` to plot the solution \(y(t)\) over the interval \([0, 64\pi]\). This will give a visual representation of how \(y(t)\) behaves over the specified time range.

Key concepts

Second-Order Linear Differential Equation

A second-order linear differential equation involves derivatives up to the second order of the unknown function. These equations are typically in the form \( ay'' + by' + cy = g(t) \), where \( a, b, c \) are constants and \( g(t) \) is a given function. These equations are crucial in modeling various physical systems, such as oscillations in mechanics and electrical circuits. Understanding them involves familiarizing oneself with the methods to solve them, often using characteristic equations, particular solutions, and sometimes numerical methods.

Initial Conditions

Initial conditions are values given for the function and its derivatives at a specific point. In our problem, initial conditions are provided as \( y(0) = 0 \) and \( y'(0) = 0 \). These conditions are essential for determining a unique solution from the family of possible solutions to the differential equation. They literally 'pin' down the path that the solution must take.

Particular Solution

A particular solution satisfies a non-homogeneous differential equation. It is a specific answer that fits the differential equation with given initial conditions. For our exercise, the non-homogeneity comes from the term \( 3\cos(2.1t) \). The particular solution adds a specific function to the general solution that precisely cancels out the effects of this non-homogeneous term.

Plotting

Plotting is a valuable way of visually understanding the behavior of a solution over a particular interval. In this exercise, we use the function `ezplot` to visualize the solution \( y(t) \) over the interval \([0, 64\pi]\). Plotting allows one to see the oscillatory nature of the response and evaluate other qualitative features like amplitude and frequency changes.

Simplification

Simplification refers to reducing a solution to its simplest form, making it easier to understand and apply. Software tools like MATLAB have built-in capabilities, such as the `simple` function, which help to reduce complex expressions to more workable forms. This process ensures that no unnecessary terms clutter the solution, thus providing a clearer interpretation.

MATLAB

MATLAB is a numerical computing environment often used for solving differential equations among other tasks. In this context, MATLAB provides tools like `dsolve` for finding symbolic solutions, and `ezplot` for visualizing them. Its powerful computation and visualization capabilities make MATLAB invaluable for studying complex mathematical models efficiently.

Graphical Representation

Graphical representation gives a visual perspective of the solution over a specified range. Often, a graph can reveal information that may not be obvious from the algebraic form. For the differential equation \( y'' + 4y = 3\cos(2.1t) \), plotting its solution helps in understanding how the function behaves over time regarding oscillations and amplitudes.

Boundary Value Problems

Boundary value problems are differential equations with conditions specified at different points in the interval of interest. While our problem here is an initial value problem, understanding boundary value conditions can be equally important. These conditions can influence how we tackle different differential equations and the kind of solutions we derive from them.

Homogeneous vs. Non-Homogeneous Equations

Homogeneous equations are those where \( g(t) = 0 \), making them simpler to solve because the solution relates only to the structure of the equation itself. Non-homogeneous equations have a \( g(t) \) term that introduces additional complexity. In our case, the equation is non-homogeneous due to the \( 3\cos(2.1t) \) term, which means we must find both the homogeneous and particular solutions.

Ordinary Differential Equations

Ordinary differential equations (ODEs) involve derivatives with respect to a single variable, usually time or space. They are distinguished from partial differential equations, which involve multiple variables. This exercise requires solving an ODE with respect to time, making it an important step in understanding dynamic systems where changes occur over time.