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Chapter 2: Problem 34

In Exercises \(34-41\), find the general solution of the differential equation. Then plot the family of solutions with the indicated initial values over the specified interval. We will use MATLAB notation to indicate the range of initial values. You can use the method of Example 7, but think about using a for loop. \(y^{\prime}+y=\sin t\) on the interval \([0,4 \pi]\) with initial conditions \(y(0)=-10: 2: 10\).

Short Answer

Expert verified
The general solution is \( y = -0.5(\sin t - \cos t) + Ce^{-t} \).

Step by step solution

01

Identify and rearrange the differential equation

The given differential equation is a first-order linear ODE: \( y' + y = \sin t \). We can express it in the standard form \( y' + p(t)y = g(t) \) where \( p(t) = 1 \) and \( g(t) = \sin t \).
02

Find the integrating factor

The integrating factor (IF) for a differential equation of the form \( y' + p(t)y = g(t) \) is calculated as \( \mu(t) = e^{\int p(t) \, dt} \). Here, \( p(t) = 1 \), so \( \mu(t) = e^{\int 1 \, dt} = e^{t} \).
03

Multiply the equation by the integrating factor

Multiply the entire equation \( y' + y = \sin t \) by the integrating factor \( e^{t} \) to get: \( e^{t}y' + e^{t}y = e^{t}\sin t \).
04

Rewrite the left side as a derivative

Notice that \( e^{t}y' + e^{t}y \) is the derivative of \( e^{t}y \) with respect to \( t \). So we rewrite the equation as: \( \frac{d}{dt}(e^{t}y) = e^{t}\sin t \).
05

Integrate both sides to find the general solution

Integrate both sides with respect to \( t \): \( \int \frac{d}{dt}(e^{t}y)\, dt = \int e^{t}\sin t \, dt \). The left hand side integrates to \( e^{t}y \). For the right hand side, use integration by parts to find \( \int e^{t}\sin t \, dt = -0.5e^{t}(\sin t - \cos t) + C \). So, \( e^{t}y = -0.5e^{t}(\sin t - \cos t) + C \).
06

Solve for y

Divide both sides by \( e^{t} \) to solve for \( y \): \( y = -0.5(\sin t - \cos t) + Ce^{-t} \). This is the general solution.
07

Plot the family of solutions

Using MATLAB, plot the solutions from \( t = 0 \) to \( t = 4\pi \) with initial conditions \( y(0) = -10: 2: 10 \). For each initial condition, solve for \( C \) using the known \( y(0) \) value and plot the corresponding solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

General Solution
The general solution to a differential equation provides a family of functions that satisfies the equation. This family of solutions is generally defined with a constant, often symbolized as \( C \). This constant allows the solution to accommodate initial conditions that may be specified in the problem.

For the given differential equation \( y' + y = \sin t \), we have derived its general solution as:
  • \( y = -0.5(\sin t - \cos t) + Ce^{-t} \)
This solution encompasses all possible solutions based on different initial conditions. In particular, \( C \) is adjusted based on the specified starting values, resulting in different particular solutions.

Each particular solution can be visualized as a curve on a graph, representing how a specific initial condition influences the behavior of the function over a given interval.
Differential Equations
Differential equations are mathematical equations that involve functions and their derivatives. They express relationships between varying quantities and are essential in modeling real-world phenomena in physics, engineering, and other disciplines.

In this exercise, we dealt with a first-order linear differential equation. These types of equations are characterized by the highest derivative being a first derivative and having a linear combination of variables and their derivatives.
  • The standard form for such an equation is: \( y' + p(t)y = g(t) \)
This form helps identify important components needed to solve the equation, like the integrating factor, which assists in reorganizing the equation into a more manageable form. Mastering these differential equations is critical for analyzing dynamic systems.
MATLAB Programming
MATLAB is an excellent tool for analyzing and visualizing differential equations. It allows users to plot solutions over specified intervals, making it easier to understand and communicate results.

In our exercise, MATLAB is used to plot the family of solutions. By leveraging MATLAB, you can:
  • Define the range of initial conditions.
  • Use loops to apply each initial condition to the general solution.
  • Visualize how these conditions affect the solution over the interval \([0, 4\pi]\).
MATLAB鈥檚 powerful features make it easier to manage complex computations and produce clear graphical representations, which is crucial for interpreting results and understanding how systems behave under different scenarios.
Integration by Parts
Integration by parts is a technique used in calculus to simplify the process of finding the integral of a product of functions. This method is particularly useful in solving integrals that are not straightforward.

The integration by parts formula is:
  • \( \int u \, dv = uv - \int v \, du \)
In our solution, integration by parts was essential in dealing with the right-hand side integral \( \int e^t\sin t \, dt \). By choosing appropriate \( u \) and \( dv \), we could simplify the integral step by step, eventually finding:
  • \( -0.5e^t(\sin t - \cos t) + C \)
This method breaks down the problem, making it possible to manage and calculate complex integrals that appear in differential equations.

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Most popular questions from this chapter

If the Symbolic Toolbox is installed in your MATLAB system, use the dsolve command to find the solution of the first order initial value problems in Exercises 15 - 18. Use the ezplot command to plot the solution over the indicated interval. y^{\prime}=-2 t y, y(0)=1,[-2,2]

For each set of parametric equations in Exercises \(29-32\), use a script file to create two plots. First, draw a plot of both \(x\) and \(y\) versus \(t\). Use handle graphics to apply different linestyles and color to the plots of \(x\) and \(y\), then add a legend, axis labels, and a title. Open a second figure window by placing the \(f\) igure command at this point in your script, then draw a plot of \(y\) versus \(x\). Add axis labels and a title to your second plot. \(x=\cos (2 t)-8 \sin (2 t), y=-5 \sin (2 t)-\cos (2 t),[0,4 \pi]\)

In Exercises \(1-6\), find the solution to the indicated initial value problem, and use ezplot to plot it. \(y^{\prime}=-t y\) with \(y(0)=1\) over \([0,2]\).

Use the appropriate sum-to-product identity from trigonometry to show that $$ \sin (12 t)-\sin (14 t)=-2 \sin (t) \cos (13 t) $$ On one plot, plot \(y=-2 \sin (t) \cos (13 t)\) and its envelopes \(y=\pm 2 \sin (t)\) over the time interval \([-2 \pi, 2 \pi]\). Use the selection tool on the figure toolbar to select each envelope, then right-click the selected envelope to change both its color and linewidth.

If the Symbolic Toolbox is installed in your MATLAB system, use the dsolve command to find the solution of the first order initial value problems in Exercises 15 - 18. Use the ezplot command to plot the solution over the indicated interval. \(y^{\prime}+2 y=\cos (t), y(0)=1,[0,20]\)
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