Question

In each exercise, assume that \(y_{1}\) and \(y_{2}\) are solutions of \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\), where \(p(t)\) and \(q(t)\) are continuous on \((a, b)\). Explain why \(y_{1}(t)\) and \(y_{2}(t)\) cannot form a fundamental set of solutions.\(y_{1}(t)\) and \(y_{2}(t)\) have a common zero in \((a, b) ;\) that is, \(y_{1}\left(t_{0}\right)=0\) and \(y_{2}\left(t_{0}\right)=0\) at some point \(t_{0}\) in \((a, b)\).

Short answer

To sum up, the functions \(y_1(t)\) and \(y_2(t)\) cannot form a fundamental set of solutions for the given differential equation because they are linearly dependent. This can be shown by evaluating the Wronskian at a common zero, where the Wronskian is proven to be zero, indicating linear dependence.

Step-by-step solution

Define Linear Independence and Dependence

Linear independence and dependence are properties of functions that describe their relationship with one another. A set of functions is linearly independent if none of the functions can be expressed as a linear combination of the other. Conversely, if any of the functions can be expressed as a linear combination of the others, the set is linearly dependent. For our given functions \(y_1(t)\) and \(y_2(t)\), we must show that they are linearly dependent to prove they cannot form a fundamental set of solutions.

Establish the Wronskian

The Wronskian, denoted by \(W(y_1, y_2)\), is a determinant used to test for linear independence of two solutions. If the Wronskian is nonzero, the solutions are linearly independent. If it is zero, the solutions are linearly dependent. The Wronskian for our two solutions \(y_1(t)\) and \(y_2(t)\) is given by: \[ W(y_1, y_2) = \begin{vmatrix} y_1(t) & y_2(t) \\ y_1'(t) & y_2'(t) \end{vmatrix} = y_1(t)y_2'(t) - y_1'(t)y_2(t) \]

Use Given Condition to Show Linear Dependence

We are given that \(y_1(t_0) = 0\) and \(y_2(t_0) = 0\) for some point \(t_0\) in \((a, b)\). Let's examine the Wronskian at \(t_0\): \[ W(y_1, y_2)(t_0) = y_1(t_0)y_2'(t_0) - y_1'(t_0)y_2(t_0) = 0\cdot y_2'(t_0) - y_1'(t_0)\cdot 0 = 0 \] Now it has been shown that the Wronskian of \(y_1(t)\) and \(y_2(t)\) is zero at some point \(t_0\) in \((a, b)\). Thus, we can conclude that \(y_1(t)\) and \(y_2(t)\) are linearly dependent.

Conclusion

Since \(y_1(t)\) and \(y_2(t)\) are linearly dependent, they cannot form a fundamental set of solutions for the given differential equation.

Key concepts

Linear Independence

In the context of differential equations, understanding linear independence is crucial. When we talk about functions being linearly independent, we mean that no function in the set can be written as a linear combination of the others. Linear combinations involve some coefficients multiplying each function, and if the only way to satisfy this equality is for all coefficients to be zero, the functions are independent.

Consider two functions, say, \( y_1(t) \) and \( y_2(t) \): if neither can be expressed as a scalar multiple of the other, then they are independent. If, however, you find that \( y_2(t) = c \cdot y_1(t) \) for some constant \( c \), they are dependent. This concept is a foundation for constructing solutions to differential equations.

• Independent functions provide distinct information about solutions.
• Dependent functions could essentially repeat the same information.

Wronskian

The Wronskian is a practical tool for checking the linear independence of functions. Named after the mathematician Józef Wroński, the Wronskian of two functions \( y_1(t) \) and \( y_2(t) \) is defined as the determinant:\[ W(y_1, y_2) = y_1(t)y_2'(t) - y_1'(t)y_2(t) \]

The key idea behind the Wronskian is pretty intuitive; if this determinant is non-zero at some point in the interval, the functions are linearly independent within that interval. A zero Wronskian at some points doesn't immediately infer dependency, but if it is zero throughout the interval, the functions are dependent.

• A non-zero Wronskian indicates independence at specific points.
• A zero Wronskian may suggest dependence, especially if it holds throughout.

Fundamental Set of Solutions

In solving homogeneous differential equations, the concept of a fundamental set of solutions is significant. This term refers to a set of solutions that can be combined linearly to express the general solution of the differential equation. For a second-order differential equation, a fundamental set typically consists of two linearly independent solutions.

The reason we emphasize linear independence is that independent solutions ensure coverage of all possible solutions. If solutions are dependent, we risk missing out on the full set of possible solutions. A fundamental set serves as the building blocks for constructing any specific solution.

• A complete fundamental set covers the whole solution space.
• Linear independence within this set is crucial for full coverage.

Homogeneous Differential Equation

Homogeneous differential equations form the backbone of many analytical and numerical studies in differential equations. These equations are characterized by an equality where all terms involve the dependent variable and its derivatives, with zero on the other side, like \( y'' + p(t)y' + q(t)y = 0 \).

The term 'homogeneous' here implies a kind of uniformity or balance, where all effects cancel each other out in a way that the solution can settle to zero. The solutions to these equations, especially when concerning their independence, can couple with concepts like linear dependence and the Wronskian to explain phenomena or to solve initial-value problems.

• They contain only derivatives and functions of the dependent variable.
• Simplified form: leads to structured, elegant solutions.