Problem 10 - Advanced Engineering Mathematics

Question

Without the aid of the Wronskian determine whether the given set of functions is linearly independent or linearly dependent on the indicated interval. (a) \(f_{1}(x)=\ln x, f_{2}(x)=\ln x^{2},(0, \infty)\) (b) \(f_{1}(x)=x^{n}, f_{2}(x)=x^{n+1}, n=1,2, \ldots,(-\infty, \infty)\) (c) \(f_{1}(x)=x, f_{2}(x)=x+1,(-\infty, \infty)\) (d) \(f_{1}(x)=\cos (x+\pi / 2), f_{2}(x)=\sin x,(-\infty, \infty)\) (e) \(f_{1}(x)=0, f_{2}(x)=x,(-5,5)\) (f) \(f_{1}(x)=2, f_{2}(x)=2 x,(-\infty, \infty)\) (g) \(f_{1}(x)=x^{2}, f_{2}(x)=1-x^{2}, f_{3}(x)=2+x^{2},(-\infty, \infty)\) (h) \(f_{1}(x)=x e^{x+1}, f_{2}(x)=(4 x-5) e^{x}, f_{3}(x)=x e^{x},(-\infty, \infty)\)

Step-by-Step Solution

Answer

(a) Dependent, (b) Independent, (c) Independent, (d) Dependent, (e) Dependent, (f) Independent, (g) Dependent, (h) Dependent.

Step 1: Analyze Functions in Part (a)

Consider the functions \(f_1(x) = \ln(x)\) and \(f_2(x) = \ln(x^2)\). Notice that \(f_2(x)\) can be rewritten as \(f_2(x) = 2\ln(x) = 2f_1(x)\). This shows that \(f_2(x)\) is a scalar multiple of \(f_1(x)\). Thus, the functions are linearly dependent on the interval \((0, \infty)\).

Step 2: Analyze Functions in Part (b)

Consider the functions \(f_1(x) = x^n\) and \(f_2(x) = x^{n+1}\). Suppose there exist constants \(c_1\) and \(c_2\) such that \(c_1x^n + c_2x^{n+1} = 0\) for all \(x\). Factoring out \(x^n\), we get \(x^n(c_1 + c_2x) = 0\). For this to be true for all \(x\), \(c_1\) and \(c_2\) must both be zero. Therefore, the functions are linearly independent on \((-\infty, \infty)\).

Step 3: Analyze Functions in Part (c)

Consider the functions \(f_1(x) = x\) and \(f_2(x) = x + 1\). Suppose there exist constants \(c_1\) and \(c_2\) such that \(c_1x + c_2(x+1) = 0\) for all \(x\). This simplifies to \((c_1 + c_2)x + c_2 = 0\). Equating coefficients, we get \(c_1 + c_2 = 0\) and \(c_2 = 0\), which implies \(c_1 = 0\). Thus, the functions are linearly independent on \((-\infty, \infty)\).

Step 4: Analyze Functions in Part (d)

Consider the functions \(f_1(x) = \cos(x + \pi/2)\) and \(f_2(x) = \sin(x)\). Using the trigonometric identity \(\cos(x + \pi/2) = -\sin(x)\), we see that \(f_1(x) = -f_2(x)\). This shows that \(f_1(x)\) is a scalar multiple of \(f_2(x)\). Thus, the functions are linearly dependent on \((-\infty, \infty)\).

Step 5: Analyze Functions in Part (e)

Consider the functions \(f_1(x) = 0\) and \(f_2(x) = x\). The zero function \(f_1(x)\) is trivially dependent on \(f_2(x)\) since it can be represented as \(0 \cdot f_2(x)\). Therefore, these functions are linearly dependent on \((-5, 5)\).

Step 6: Analyze Functions in Part (f)

Consider the functions \(f_1(x) = 2\) and \(f_2(x) = 2x\). Suppose there exist constants \(c_1\) and \(c_2\) such that \(c_1 \cdot 2 + c_2 \cdot 2x = 0\) for all \(x\). This simplifies to \(2(c_1 + c_2x) = 0\). For this to be zero for all \(x\), \(c_1\) must equal zero and \(c_2\) as well, if \(xeq 0\). Thus, these functions are linearly independent on \((-\infty, \infty)\).

Step 7: Analyze Functions in Part (g)

Consider the functions \(f_1(x) = x^2\), \(f_2(x) = 1 - x^2\), and \(f_3(x) = 2 + x^2\). We form the linear combination \(c_1x^2 + c_2(1-x^2) + c_3(2+x^2) = 0\). This simplifies to \((c_1 - c_2 + c_3)x^2 + c_2 + 2c_3 = 0\). Solving the coefficients gives two equations: \(c_1 - c_2 + c_3 = 0\) and \(c_2 + 2c_3 = 0\). This system has a non-trivial solution, implying linear dependence on \((-\infty, \infty)\).

Step 8: Analyze Functions in Part (h)

Consider the functions \(f_1(x) = xe^{x+1}\), \(f_2(x) = (4x-5)e^x\), and \(f_3(x) = xe^x\). Simplifying, \(f_1(x) = xe \cdot e^x\) and \(f_3(x) = xe^x\), so \(f_1(x) = e \cdot f_3(x)\). This shows \(f_1(x)\) is a scalar multiple of \(f_3(x)\), indicating linear dependence among these functions on \((-\infty, \infty)\).

Key Concepts

Linear Dependence

Linear dependence is a fundamental concept in linear algebra. When we say a set of functions is linearly dependent, it means that at least one of the functions can be expressed as a linear combination of the others. In simpler terms, one function in the set does not add any new information because it is simply a scaled version or an algebraic mix of the others.

• For example, if you have functions like \(f_1(x) = \ln(x)\) and \(f_2(x) = \ln(x^2)\), understanding that \(f_2(x)\) can be rewritten as \(2f_1(x)\) shows they are dependent.
• This dependency concept can extend to any scenario where one set of functions can mimic another completely via multiplication or addition.

Recognizing linear dependence allows mathematicians to simplify problems, reducing the number of dimensions they have to consider, leading to more efficient solutions.

Function Analysis

Function analysis involves scrutinizing functions to understand their properties and behaviors. When we analyze sets of functions, especially when determining linear independence or dependence, we look for relationships between them.

• For instance, in the case of \(f_1(x) = x^n\) and \(f_2(x) = x^{n+1}\), by attempting to express one function as a combination like \(c_1x^n + c_2x^{n+1}\), function analysis helps us determine if such a combination is always zero, signaling independence.
• Analyzing functions might involve algebraic manipulation or finding scalar multiples or zero solutions that prove linear dependence or independence.

In essence, function analysis is a core toolset in solving and simplifying mathematical problems, guiding mathematicians in establishing key relationships among functions.

Interval Analysis

Interval analysis is the study of the behavior of functions over specific intervals on the real number line. When determining the linear independence or dependence of functions, the interval can sometimes play a crucial role.

• For example, the linear dependence of \(f_1(x) = 0\) and \(f_2(x) = x\) on the interval \((-5, 5)\) is straightforward as the zero function is universally dependent.
• However, the interval may affect results when working with functions involving division or undefined points, although in many cases, the conclusions regarding linear independence or dependence might remain consistent across different intervals.

In every function analysis task, especially those involving dependence or independence, thoroughly understanding the interval implications ensures accurate and meaningful conclusions.