Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 3: Problem 26

verify that the functions \(y_{1}\) and \(y_{2}\) are solutions of the given differential equation. Do they constitute a fundamental set of solutions? $$ (1-x \cot x) y^{\prime \prime}-x y^{\prime}+y=0, \quad 0

Short Answer

Expert verified
Based on the provided step-by-step solution, we can create the following short answer: Both \(y_1(x) = x\) and \(y_2(x) = \sin x\) are solutions to the given differential equation \((1-x \cot x) y^{\prime\prime}-x y^{\prime}+y=0\). The Wronskian, \(W(y_1, y_2) = x\cos x - \sin x\), is not identically zero in the domain \(0 < x < \pi\). Therefore, \(y_1(x) = x\) and \(y_2(x) = \sin x\) form a fundamental set of solutions to the given differential equation.

Step by step solution

Achieve better grades quicker with Premium

  • Unlimited AI interaction
  • Study offline
  • Say goodbye to ads
  • Export flashcards
Start your free trial Skip

Over 22 million students worldwide already upgrade their learning with Vaia!

01

Analyze the given differential equation

The given differential equation is \((1-x \cot x) y^{\prime\prime}-x y^{\prime}+y=0\). Our task is to verify if \(y_1(x) = x\) and \(y_2(x) = \sin x\) are solutions to this differential equation.
02

Substitute \(y_1\) and check if it satisfies the differential equation

Let's take \(y_1(x) = x\) and calculate its first and second derivatives: \(y_1^{\prime}(x) = 1\), \(y_1^{\prime\prime}(x)= 0\). Now, let's substitute \(y_1(x)\), \(y_1^{\prime}(x)\), and \(y_1^{\prime\prime}(x)\) into the differential equation: \((1-x \cot x)(0)-x(1)+x=0\). Simplify the expression: \(x - x = 0\). This is a true statement, so \(y_1(x) = x\) is a solution to the given differential equation.
03

Substitute \(y_2\) and check if it satisfies the differential equation

Let's take \(y_2(x) = \sin x\) and calculate its first and second derivatives: \(y_2^{\prime}(x) = \cos x\), \(y_2^{\prime\prime}(x)= -\sin x\). Now, let's substitute \(y_2(x)\), \(y_2^{\prime}(x)\), and \(y_2^{\prime\prime}(x)\) into the differential equation: \((1-x \cot x)(-\sin x)-x(\cos x)+\sin x=0\). Multiply and simplify the expression: \(-\sin x + x\sin x + x\cos x + \sin x = 0\). This is a true statement, so \(y_2(x) = \sin x\) is also a solution to the given differential equation.
04

Determine if \(y_1\) and \(y_2\) form a fundamental set of solutions

To check if \(y_1(x)\) and \(y_2(x)\) form a fundamental set of solutions, we need to compute their Wronskian: \(W(y_1, y_2) = \begin{vmatrix}y_1 & y_2 \\ y_1^{\prime} & y_2^{\prime}\end{vmatrix} = \begin{vmatrix}x & \sin x \\ 1 & \cos x \end{vmatrix}\). Now, calculate the determinant: \(W(y_1, y_2) = x\cos x - \sin x\). The function \(W(y_1, y_2) = x\cos x - \sin x\) is not identically zero in the domain \(0 < x < \pi\). Thus, the given functions \(y_1(x) = x\) and \(y_2(x) = \sin x\) form a fundamental set of solutions to the given differential equation.

Key Concepts

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

Fundamental Set of Solutions
Understanding the fundamental set of solutions is critical when dealing with homogeneous linear differential equations. It's about finding a group of solutions that can represent all possible solutions. With these 'building blocks,' any other solution of the differential equation can be expressed as a linear combination of the elements in this set.

In the exercise, we analyzed if the functions \(y_1(x) = x\) and \(y_2(x) = \sin x\) are part of such a set for the differential equation \( (1-x \cot x) y^{\prime \prime}-x y^{\prime}+y=0\). After verifying that both functions satisfy the differential equation independently, it was essential to determine if they are linearly independent. This is done using the Wronskian, which we will discuss next. If the Wronskian of these functions is not zero, it indicates that they are indeed linearly independent and, therefore, constitute a fundamental set of solutions. This fundamental set provides us with a powerful tool to describe all solutions to the differential equation in the given domain \(0
Wronskian
The Wronskian is a concept named after the Polish mathematician Józef Hoene-Wronski. It is a method used to determine whether a set of functions is linearly independent, an essential aspect of finding the fundamental set of solutions. By calculating the Wronskian of two functions, we are essentially taking the determinant of a matrix composed of these functions and their derivatives.

In our exercise, we calculated the Wronskian of \(y_1\) and \(y_2\) as follows:
\begin{align*} W(y_1, y_2) &= \begin{vmatrix} y_1 & y_2 \ y_1^{\prime} & y_2^{\prime}\end{vmatrix} = \begin{vmatrix} x & \sin x \ 1 & \cos x \end{vmatrix} \ &= x\cos x - \sin x.\end{align*}
The result, \(W(y_1, y_2) = x\cos x - \sin x\), is not constantly zero in the interval \(0
Solving Differential Equations
Solving differential equations is a staple of mathematical study and application, with techniques varying widely based on the type of equation at hand. Whether dealing with ordinary differential equations (ODEs) or partial differential equations (PDEs), the solutions can describe a multitude of physical phenomena, such as sound waves, heat conduction, or population growth.

In the textbook exercise, we focused on solving a specific type of ODE, verifying given solutions, and then determining their independence using the Wronskian. The successful verification process not only confirmed our solutions but also empowered us to use the fundamental set of solutions for constructing general solutions for similar differential equations within the given domain. The ability to solve such equations and understand their solutions' structure allows us to model and predict a vast range of real-world situations—underlining the power and necessity of mastery in solving differential equations.

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 a vibrating system described by the initial value problem $$ u^{\prime \prime}+\frac{1}{4} u^{\prime}+2 u=2 \cos \omega t, \quad u(0)=0, \quad u^{\prime}(0)=2 $$ (a) Determine the steady-state part of the solution of this problem. (b) Find the amplitude \(A\) of the steady-state solution in terms of \(\omega\). (c) Plot \(A\) versus \(\omega\). (d) Find the maximum value of \(A\) and the frequency \(\omega\) for which it occurs.

Show that the solution of the initial value problem $$ L[y]=y^{\prime \prime}+p(t) y^{\prime}+q(t) y=g(t), \quad y\left(t_{0}\right)=y_{0}, \quad y^{\prime}\left(t_{0}\right)=y_{0}^{\prime} $$ can be written as \(y=u(t)+v(t)+v(t),\) where \(u\) and \(v\) are solutions of the two initial value problems $$ \begin{aligned} L[u] &=0, & u\left(t_{0}\right)=y_{0}, & u^{\prime}\left(t_{0}\right)=y_{0}^{\prime} \\ L[v] &=g(t), & v\left(t_{0}\right)=0, & v^{\prime}\left(t_{0}\right) &=0 \end{aligned} $$ respectively. In other words, the nonhomogeneities in the differential equation and in the initial conditions can be dealt with separately. Observe that \(u\) is easy to find if a fundamental set of solutions of \(L[u]=0\) is known.

Consider the initial value problem $$ u^{\prime \prime}+\gamma u^{\prime}+u=0, \quad u(0)=2, \quad u^{\prime}(0)=0 $$ We wish to explore how long a time interval is required for the solution to become "negligible" and how this interval depends on the damping coefficient \(\gamma\). To be more precise, let us seek the time \(\tau\) such that \(|u(t)|<0.01\) for all \(t>\tau .\) Note that critical damping for this problem occurs for \(\gamma=2\) (a) Let \(\gamma=0.25\) and determine \(\tau,\) or at least estimate it fairly accurately from a plot of the solution. (b) Repeat part (a) for several other values of \(\gamma\) in the interval \(0<\gamma<1.5 .\) Note that \(\tau\) steadily decreases as \(\gamma\) increases for \(\gamma\) in this range. (c) Obtain a graph of \(\tau\) versus \(\gamma\) by plotting the pairs of values found in parts (a) and (b). Is the graph a smooth curve? (d) Repeat part (b) for values of \(\gamma\) between 1.5 and \(2 .\) Show that \(\tau\) continues to decrease until \(\gamma\) reaches a certain critical value \(\gamma_{0}\), after which \(\tau\) increases. Find \(\gamma_{0}\) and the corresponding minimum value of \(\tau\) to two decimal places. (e) Another way to proceed is to write the solution of the initial value problem in the form (26). Neglect the cosine factor and consider only the exponential factor and the amplitude \(R\). Then find an expression for \(\tau\) as a function of \(\gamma\). Compare the approximate results obtained in this way with the values determined in parts (a), (b), and (d).

The position of a certain spring-mass system satisfies the initial value problem $$ u^{\prime \prime}+\frac{1}{4} u^{\prime}+2 u=0, \quad u(0)=0, \quad u^{\prime}(0)=2 $$ (a) Find the solution of this initial value problem. (b) Plot \(u\) versus \(t\) and \(u^{\prime}\) versus \(t\) on the same axes. (c) Plot \(u\) versus \(u\) in the phase plane (see Problem 28 ). Identify several corresponding points on the curves in parts (b) and (c). What is the direction of motion on the phase plot as \(t\) increases?

In the absence of damping the motion of a spring-mass system satisfies the initial value problem $$ m u^{\prime \prime}+k u=0, \quad u(0)=a, \quad u^{\prime}(0)=b $$ (a) Show that the kinetic energy initially imparted to the mass is \(m b^{2} / 2\) and that the potential energy initially stored in the spring is \(k a^{2} / 2,\) so that initially the total energy in the system is \(\left(k a^{2}+m b^{2}\right) / 2\). (b) Solve the given initial value problem. (c) Using the solution in part (b), determine the total energy in the system at any time \(t .\) Your result should confirm the principle of conservation of energy for this system.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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