Question

Assume that \(p\) and \(q\) are continuous, and that the functions \(y_{1}\) and \(y_{2}\) are solutions of the differential equation \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\) on an open interval \(I\) Prove that if \(y_{1}\) and \(y_{2}\) have a common point of inflection \(t_{0}\) in \(I,\) they cannot be a fundamental set of solutions on \(I\) unless both \(p\) and \(q\) are zero at \(t_{0} .\)

Short answer

Question: Prove that if \(y_1\) and \(y_2\) are solutions of the linear homogeneous second-order differential equation \(y''+p(t)y'+q(t)y=0\), where \(p\) and \(q\) are continuous functions on an interval I, and both have a common point of inflection at \(t_0\), then \(y_1\) and \(y_2\) cannot be a fundamental set of solutions on I unless both \(p(t_0)\) and \(q(t_0)\) are equal to zero.

Step-by-step solution

Recall the definition of points of inflection

A point of inflection occurs when the second derivative of a function changes its sign, i.e., from positive to negative or from negative to positive. Since \(y_1\) and \(y_2\) have a common point of inflection at \(t_0\), their second derivatives, \(y_1''(t_0)\) and \(y_2''(t_0)\), are zero.

Substitute the given functions in the differential equation

We are given the differential equation \(y'' + p(t) y' + q(t) y = 0\). Substitute \(y_1\) into the equation and set \(t = t_0\): \(y_1''(t_0) + p(t_0)y_1'(t_0) + q(t_0)y_1(t_0) = 0\) Similarly, substitute \(y_2\) into the equation and set \(t = t_0\): \(y_2''(t_0) + p(t_0)y_2'(t_0) + q(t_0)y_2(t_0) = 0\) Since \(y_1''(t_0)==\)y_2''(t_0) = \(0\) (from Step 1), the above equations become: \(p(t_0)y_1'(t_0) + q(t_0)y_1(t_0) = 0\) \(p(t_0)y_2'(t_0) + q(t_0)y_2(t_0) = 0\)

Check for linear independence

To be a fundamental set of solutions, \(y_1\) and \(y_2\) must be linearly independent. To check for linear independence, we can use the Wronskian determinant, denoted as \(W(y_1, y_2)\). Recall the definition of the Wronskian determinant: \(W(y_1, y_2) = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1 y_2' - y_2 y_1'\) If \(W(y_1, y_2) \neq 0\), then \(y_1\) and \(y_2\) are linearly independent.

Compute the Wronskian determinant at \(t_0\)

In this step, we need to compute the Wronskian determinant at \(t_0\): \(W(y_1, y_2)(t_0) = y_1(t_0) y_2'(t_0) - y_2(t_0) y_1'(t_0)\) Now, we can rewrite the equations from Step 2 as: \(y_1'(t_0) = -\frac{q(t_0)}{p(t_0)} y_1(t_0)\) \(y_2'(t_0) = -\frac{q(t_0)}{p(t_0)} y_2(t_0)\) Substitute these equations into the Wronskian determinant: \(W(y_1, y_2)(t_0) = y_1(t_0) \left(-\frac{q(t_0)}{p(t_0)} y_2(t_0)\right) - y_2(t_0) \left(-\frac{q(t_0)}{p(t_0)} y_1(t_0)\right) = 0\)

Conclude the proof

Since the Wronskian determinant is zero at \(t_0\), it means that \(y_1\) and \(y_2\) are linearly dependent, and thus they cannot be a fundamental set of solutions on I. We conclude that both \(p(t_0)\) and \(q(t_0)\) must be equal to zero in order to satisfy the conditions for \(y_1\) and \(y_2\) to be a fundamental set of solutions.

Key concepts

Point of Inflection

Understanding the concept of a point of inflection is critical when dealing with the behavior of functions, especially when analyzing graphs of their derivatives. A point of inflection is where a curve changes its curvature, shifting from concave upward to downward, or vice versa. Mathematically, this is where the second derivative of a function switches signs. In terms of differential equations, if both solutions have a point of inflection at the same value of the independent variable, it reveals important information about the functions and potentially their relationship to each other.

This concept is significant because it marks the change in the dynamic behavior of solutions to differential equations. In the context of solving our original equation, both solutions having a common point of inflection at t₀ indicates a pivotal behavior that leads to a deeper inquiry into the nature of the differential equation and its coefficients.

Fundamental Set of Solutions

A Fundamental Set of Solutions pertains to a set of solutions to a homogeneous linear differential equation which can be used to generate all possible solutions of that differential equation. In simpler terms, if we have two solutions of a second order homogeneous differential equation, any other solution can be constructed as a linear combination of these two. This concept is closely tied with mathematical concepts of linear independence and the Wronskian determinant.

A fundamental set provides the building blocks for the general solution of the differential equation. To be considered fundamental, the solutions must be linearly independent, which means that one solution cannot be obtained by merely multiplying the other by some scalar function. For students, recognizing and verifying a fundamental set of solutions is key to solving many differential equations successfully.

Wronskian Determinant

The Wronskian determinant is an essential tool in the study of differential equations. The Wronskian is used to establish the linear independence of functions, which in turn is critical when determining the fundamental set of solutions for differential equations. It is named after the Polish mathematician Józef Hoene-Wroński.

The Wronskian of two functions, denoted by W(y₁, y₂), is a determinant that combines both functions and their first derivatives. It is calculated as follows:
\[W(y_1, y_2) = \begin{vmatrix} y_1 & y_2 \ y_1' & y_2' \end{vmatrix} = y_1y_2' - y_2y_1'\]
Your ability to compute and interpret the Wronskian is crucial, as a non-zero Wronskian at any point in the interval of interest implies that the set of functions are linearly independent, thus forming a fundamental set. However, a zero Wronskian doesn't always mean dependence and additional checks may be necessary.

Linear Independence

The notion of linear independence is a cornerstone in the study of differential equations and vector space theory. Two or more functions are said to be linearly independent if no function in the set can be written as a linear combination of the others. For instance, in the context of our differential equation, solutions y₁ and y₂ are considered linearly independent if there are no constants c₁ and c₂ (not both zero) such that c₁y₁ + c₂y₂ = 0 for all t in the interval I.

Understanding and identifying linear independence is vital for students, as it is directly related to the uniqueness of solutions of differential equations. If the set of solutions were dependent, they would not span the solution space sufficiently to describe all possible behavior of the equation under study. Linear independence ensures that the solution set is as 'wide' as needed to include all possible scenarios.