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Chapter 10: Problem 48

For each of the following sets of linearly dependent functions find \(k_{1}, k_{2}, \ldots\) such that \(k_{1} f_{1}+k_{2} f_{2}+\ldots=0\) (a) \(\\{t+1, t, 2\\}\) (b) \(\left\\{t^{2}-1, t^{2}+1, t-1, t+1\right\\}\)

Short Answer

Expert verified
(a) The coefficients are \(k_1 = 1\), \(k_2 = -1\), \(k_3 = -\frac{1}{2}\). (b) The coefficients are \(k_1 = 1\), \(k_2 = -1\), \(k_3 = -1\), \(k_4 = 1\).

Step by step solution

01

Identify the dependence in Set (a)

The functions given are \(f_1 = t + 1\), \(f_2 = t\), and \(f_3 = 2\). To find \(k_1, k_2, k_3\) such that \(k_1 (t+1) + k_2 t + k_3 \times 2 = 0\), we first express the equation: \(k_1 t + k_1 + k_2 t + 2k_3 = 0\).
02

Equating coefficients from Set (a)

Combine like terms to get \((k_1 + k_2) t + (k_1 + 2k_3) = 0\). For the expression to be zero for all \(t\), the coefficients of \(t\) and the constant term both need to be zero: \(k_1 + k_2 = 0\) and \(k_1 + 2k_3 = 0\).
03

Solve the equations for Set (a)

From \(k_1 + k_2 = 0\), we get \(k_2 = -k_1\). Substitute \(k_2 = -k_1\) into the second equation: \(k_1 + 2k_3 = 0\). Thus, \(2k_3 = -k_1\) or \(k_3 = -\frac{1}{2}k_1\). Choose \(k_1 = 1\) for simplicity, then \(k_2 = -1\) and \(k_3 = -\frac{1}{2}\).
04

Identify dependence in Set (b)

The functions are \(f_1 = t^2 - 1\), \(f_2 = t^2 + 1\), \(f_3 = t - 1\), \(f_4 = t + 1\). We need \(k_1 (t^2 - 1) + k_2 (t^2 + 1) + k_3 (t - 1) + k_4 (t + 1) = 0\), leading to \((k_1 + k_2) t^2 + (k_3 + k_4) t + (-k_1 + k_2 - k_3 + k_4) = 0\).
05

Equating coefficients from Set (b)

We equate real coefficients: \(k_1 + k_2 = 0\), \(k_3 + k_4 = 0\), and \(-k_1 + k_2 - k_3 + k_4 = 0\).
06

Solve the equations for Set (b)

From \(k_1 + k_2 = 0\), we get \(k_2 = -k_1\). From \(k_3 + k_4 = 0\), we get \(k_4 = -k_3\). Substituting \(k_2 = -k_1\) and \(k_4 = -k_3\) into the third equation, \(-k_1 - k_1 - k_3 - k_3 = 0\), implies \(-2k_1 - 2k_3 = 0\). Thus, \(k_1 = -k_3\). By choosing \(k_1 = 1\), \(k_2 = -1\), \(k_3 = -1\), and \(k_4 = 1\), we satisfy all the equations.

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Key Concepts

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

Systems of Equations
When dealing with systems of equations, often we try to find solutions that satisfy multiple equations at the same time. In the context of linear dependence, we focus on whether there is a combination of coefficients (like \(k_1, k_2, ...\)) that can make a linear combination of given functions equal zero. This is essentially testing whether the functions can provide unique solutions or if they overlap.
  • A system is composed of multiple equations that we solve together.
  • In linear dependence, we want the entire expression to equate to zero, meaning there's overlap.
  • Given functions need coefficients that will satisfy the zero condition for all values of \(t\).
For the exercise, solving involves equating the coefficients of corresponding terms (\(t^n\) terms and constants) from both sides of the equation to zero. It leads to a set of linear equations that reveal the relationships among the coefficients. This helps shows linear dependence.
Polynomial Functions
Polynomial functions are expressions involving powers of a variable, like \(t\) in the exercises provided. They come in different degrees depending on the highest power. Understanding their properties helps in figuring out the coefficients which show dependency.
  • The degree of a polynomial is determined by its highest power, which influences its shape and solutions.
  • Each term in a polynomial function has a coefficient that plays a significant role in determining its value.
  • In linear combination exercises, identifying like terms is crucial.
By comparing polynomial terms in the step-by-step solution, we align terms with the same degree and solve accordingly. This is foundational, as like terms contribute to whether a function or set of functions is dependent or independent.
Matrix Theory
Matrix theory provides a structured way to organize and solve systems of linear equations. Matrices consist of rows and columns that can represent coefficients from a system and be manipulated to solve for unknowns.
  • Matrices are powerful tools in solving linear systems, facilitating multiple equation solutions simultaneously.
  • In our context, the coefficients \(k_1, k_2, ...\) act as the unknowns.
  • Operations like matrix multiplication and row reduction help find solutions efficiently.
In the exercise context, while we solve using coefficients' alignment, matrix methods like Gaussian elimination or Cramer's rule can be applied for more complex scenarios. This adds a layer of understanding to the structure and solutions to problems involving multiple equations and dependency.

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Most popular questions from this chapter

Find the solution of the following initial-value problem: $$ x^{3} t \frac{\mathrm{d} x}{\mathrm{~d} t}=t^{4}+x^{4}, \quad x(1)=4 $$

Find the value of \(X(1)\) for the initial-value problem $$ \frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{x}{2 \sqrt{(} t+x)}, \quad x(0.5)=1 $$ using Euler's method with step size \(h=0.1\).

Find the damping parameters and natural frequencies of the systems governed by the following second-order linear constant-coefficient differential equations: (a) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+6 \frac{\mathrm{d} x}{\mathrm{~d} t}+9 x=0\) (b) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+4 \frac{\mathrm{d} x}{\mathrm{~d} t}+7 x=0\)

Write a computer program to solve the initialvalue problem $$ \begin{aligned} &\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+x^{2} \frac{\mathrm{d} x}{\mathrm{~d} t}+x=\sin t \\ &x(0)=0, \quad \frac{\mathrm{d} x}{\mathrm{~d} t}(0)=1 \end{aligned} $$ using Euler's method. Use your program to find the value of \(X(0.4)\) using steps of \(h=0.01\) and \(h=0.005\). Hence estimate the accuracy of your value of \(X(0.4)\) and estimate the step size that would be necessary to obtain a value of \(X(0.4)\) accurate to \(4 \mathrm{dp}\).

For each of the following sets of linearly dependent functions find \(k_{1}, k_{2}, \ldots\) such that \(k_{1} f_{1}+k_{2} f_{2}+\ldots=0\) (a) \(\\{\sin t, \cos t+\sin t, \cos 2 t-\sin t, \cos t-\cos 2 t\\}\) (b) \(\left\\{t+t^{3}, t-t^{2}, t^{2}+2 t^{3}, t^{2}-t^{3}\right\\}\) (c) \(\left\\{\ln t, \ln 2 t, \ln 4 t^{2}\right\\}\) (d) \(\\{f(t)+g(t), f(t)(1+f(t)), g(t)-f(t)\) \(\left.f(t)^{2}-g(t)\right\\}\) (e) \(\left\\{1+t+2 t^{2}, t-2 t^{2}+3 t^{3}, 1+t-2 t^{2},\right.\), \(\left.t-2 t^{2}-3 t^{3}, t^{3}\right\\}\)
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