Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Problem 5

Consider the nonhomogeneous differential equation \(x^{\prime \prime}-3 x^{\prime}+2 x=6 e^{3 t}\). a. Find the general solution of the homogenous equation. b. Find a particular solution using the Method of Undetermined Coefficients by guessing \(x_{p}(t)=A e^{3 t}\). c. Use your answers in the previous parts to write the general solution for this problem.

Short Answer

Expert verified
The general solution of the differential equation is \(x(t) = c_1e^{2t} + c_2e^{t} + 3e^{3t}\).

Step by step solution

01

Solve the homogeneous equation

The homogeneous equation is obtained by setting the right hand side of the given equation to zero. Hence the homogeneous equation we will solve is \(x'' - 3x' + 2x = 0\). The characteristic equation of this homogeneous differential equation is \(m^2 - 3m + 2 = 0\). Factoring this equation gives us \((m-2)(m-1) = 0\). Hence, the roots are \(m = 2, 1\). This means the general solution to the homogeneous equation is \(x_h(t) = c_1e^{2t} + c_2e^{t}\), where \(c_1\) and \(c_2\) are constants to be determined by the initial conditions.
02

Guess and verify a particular solution

We now find a particular solution to the non-homogeneous equation. For the non-homogeneous term in the differential equation, which takes the form \(6e^{3t}\), we can guess a particular solution of the form \(x_p(t) = Ae^{3t}\). Plug \(x_p(t)\) back into the original differential equation and solve for the constant \(A\). After calculation, we get \(A = \frac{6}{2}=3\). So a particular solution to the non-homogeneous equation is \(x_p(t) = 3e^{3t}\).
03

Write the general solution for the problem

The general solution of a nonhomogeneous differential equation is a combination of the homogeneous solution and the particular solution. Hence the general solution is \(x(t) = x_h(t) + x_p(t) = c_1e^{2t} + c_2e^{t} + 3e^{3t}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the system $$ \begin{array}{r} x^{\prime}=-4 x-y \\ y^{\prime}=x-2 y \end{array} $$ a. Determine the second-order differential equation satisfied by \(x(t)\). b. Solve the differential equation for \(x(t)\). c. Using this solution, find \(y(t)\). d. Verify your solutions for \(x(t)\) and \(y(t)\). e. Find a particular solution to the system given the initial conditions \(x(0)=1\) and \(y(0)=0\)

Consider the case of free fall with a damping force proportional to the velocity, \(f_{D}=\pm k v\) with \(k=0.1 \mathrm{~kg} / \mathrm{s}\). a. Using the correct sign, consider a \(50-\mathrm{kg}\) mass falling from rest at a height of \(100 \mathrm{~m}\). Find the velocity as a function of time. Does the mass reach terminal velocity? b. Let the mass be thrown upward from the ground with an initial speed of \(50 \mathrm{~m} / \mathrm{s}\). Find the velocity as a function of time as it travels upward and then falls to the ground. How high does the mass get? What is its speed when it returns to the ground?

Find the general solution of each differential equation. When an initial condition is given, find the particular solution satisfying that condition. a. \(y^{\prime \prime}-3 y^{\prime}+2 y=20 e^{-2 x}, \quad y(0)=0, \quad y^{\prime}(0)=6\). b. \(y^{\prime \prime}+y=2 \sin 3 x\). c. \(y^{\prime \prime}+y=1+2 \cos x\). d. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=3 x^{2}-x, \quad x>0\).

Find the solution of each initial value problem using the appropriate initial value Green's function. a. \(y^{\prime \prime}-3 y^{\prime}+2 y=20 e^{-2 x}, \quad y(0)=0, \quad y^{\prime}(0)=6\). b. \(y^{\prime \prime}+y=2 \sin 3 x, \quad y(0)=5, \quad y^{\prime}(0)=0\). c. \(y^{\prime \prime}+y=1+2 \cos x, \quad y(0)=2, \quad y^{\prime}(0)=0\). d. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=3 x^{2}-x, \quad y(1)=\pi, \quad y^{\prime}(1)=0\).

Use i) Euler's Method and ii) the Midpoint Method to determine the given value of \(y\) for the following problems: a. \(\frac{d y}{d x}=2 y, y(0)=2\). Find \(y(1)\) with \(h=0.1\). b. \(\frac{d y}{d x}=x-y, y(0)=1\). Find \(y(2)\) with \(h=0.2\). c. \(\frac{d y}{d x}=x \sqrt{1-y^{2}}, y(1)=0 .\) Find \(y(2)\) with \(h=0.2\).
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free