Question

Suppose \(p\) and \(q\) are continuous on \((a, b)\) and \(\left\\{y_{1}, y_{2}\right\\}\) is a fundamental set of solutions of $$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$ on \((a, b) .\) Show that if \(y_{1}\left(x_{1}\right)=y_{1}\left(x_{2}\right)=0,\) where \(a

Short answer

Question: Given a second-order homogeneous linear differential equation with continuous coefficients \(p(x)\) and \(q(x)\) on the interval \((a, b)\), and a fundamental set of solutions \(\left\{y_1, y_2\right\}\), if \(y_1(x_1) = y_1(x_2) = 0\) with \(a < x_1 < x_2 < b\), prove that there exists a point \(x\) in the interval \((x_1, x_2)\) such that \(y_2(x) = 0\). Solution: In order to prove this, we showed that if \(y_2\) has no zeros in \((x_1, x_2)\), then the ratio \(y_1/y_2\) is either strictly increasing or strictly decreasing on the interval, which leads to a contradiction as \(z(x_1)=z(x_2)=0\). Therefore, there must be a point \(x \in (x_1, x_2)\) such that \(y_2(x) = 0\).

Step-by-step solution

Construct the Wronskian

Consider the Wronskian of the two solutions, \(y_1\) and \(y_2\): $$W(x) = \det \begin{pmatrix} y_1(x) & y_2(x) \\ y_1'(x) & y_2'(x) \end{pmatrix} = y_1(x)y_2'(x) - y_1'(x)y_2(x)$$ Since \(y_1\) and \(y_2\) form a fundamental set of solutions, their Wronskian is nonzero. Additionally, we know that the Wronskian of any two solutions of a homogeneous linear differential equation satisfies the Abel's identity: $$\frac{dW(x)}{dx} + p(x) W(x) = 0$$

Use the hint and find the contradiction

Suppose that there are no zeros of \(y_2\) in the interval \((x_1, x_2)\). Then, define a new function \(z(x) = \frac{y_1(x)}{y_2(x)}\). To find the derivative of z(x), set: $$z'(x) = \frac{y_1'(x)y_2(x) - y_1(x)y_2'(x)}{y_2^2(x)}$$ Now, we know that \(y_1'(x)y_2(x) - y_1(x)y_2'(x) = -W(x)\) from the definition of Wronskian, also \(W'(x) = -p(x)W(x)\) from Abel's identity, thus we have: $$z'(x) = - \frac{W(x)}{y_2^2(x)}$$ As \(W(x)\) is nonzero and continuous on our interval \((a,b)\), and \(y_2(x)\) has no zeros in the interval \((x_1, x_2)\), we see that \(z'(x)\) keeps the same sign in \((x_1, x_2)\). If \(z'(x) > 0\), then \(z(x)\) is strictly increasing. If \(z'(x) < 0\), then \(z(x)\) is strictly decreasing. Now, consider the values of \(z(x)\) at \(x_1\) and \(x_2\). As \(y_1(x_1) = y_1(x_2) = 0\), we have \(z(x_1) = z(x_2) = 0\). But this contradicts the fact that \(z(x)\) should be either strictly increasing or strictly decreasing in the interval \((x_1, x_2)\). Therefore, our assumption that \(y_2\) has no zeros in the interval \((x_1, x_2)\) is incorrect.

Conclusion

Hence, we have shown that if \(y_1(x_1) = y_1(x_2) = 0\) and \(y_1\) and \(y_2\) are solutions to the given second-order homogeneous linear differential equation, then there must be a point \(x \in (x_1, x_2)\) such that \(y_2(x) = 0\).

Key concepts

Wronskian

The Wronskian is a determinant used in the study of differential equations to check the linear independence of a set of solutions. Consider two functions, \( y_1(x) \) and \( y_2(x) \), which are solutions to a differential equation. Their Wronskian is calculated as follows:
\[W(x) = \begin{vmatrix} y_1(x) & y_2(x) \ y_1'(x) & y_2'(x) \end{vmatrix} = y_1(x)y_2'(x) - y_1'(x)y_2(x)\]
If the Wronskian \( W(x) \) is nonzero at a point or over an interval, \( y_1 \) and \( y_2 \) are linearly independent there. This is crucial, especially when dealing with solutions of differential equations, as it means these solutions can span a solution space and are not just scalar multiples of each other.
Additionally, the Wronskian satisfies a specific identity known as Abel's identity, which helps in understanding how the Wronskian changes over an interval. This identity is given by:
\[\frac{dW(x)}{dx} + p(x) W(x) = 0\]
where \( p(x) \) is a coefficient from the corresponding differential equation.
Understanding the Wronskian not only ensures linear independence but also assists in proving properties of solutions like oscillation within specific intervals.

Fundamental Set of Solutions

A fundamental set of solutions is a group of solutions to a differential equation that are linearly independent and cover the entire solution space of the equation. Consider a homogeneous linear differential equation:
\[y'' + p(x)y' + q(x)y = 0\]
Here, \( \{y_1(x), y_2(x)\} \) is a fundamental set of solutions if any other solution \( y(x) \) can be expressed as a linear combination of \( y_1(x) \) and \( y_2(x) \). This means:
\[y(x) = c_1 y_1(x) + c_2 y_2(x)\]
where \( c_1 \) and \( c_2 \) are constants.
The concept of a fundamental set is critical because it provides the basis needed to construct the general solution to the differential equation. The formation of a consistent, nonzero Wronskian between the solutions further guarantees that they form such a set, reinforcing their linear independence and completeness in representing the solution space.

Continuous Functions

Continuous functions are those whose graphs are unbroken or uninterrupted over their domain, meaning there are no gaps or jumps. In calculus and differential equations, continuity is crucial for ensuring well-defined operations like differentiation and integration.
For a given function \( f(x) \), it is continuous if for every \( x_0 \) in the domain, the limit:
\[\lim_{x \to x_0} f(x) = f(x_0)\]
In the context of differential equations, especially in the problem under discussion, functions \( p(x) \) and \( q(x) \) need to be continuous over an interval \((a, b)\). This requirement ensures that the solutions to the differential equation have properties like differentiability and fulfill specific theorems that predict behavior over the interval.
Continuous coefficients in a differential equation facilitate reliable existence and uniqueness results for its solutions, ensuring the predictable behavior of the solution set.

Homogeneous Linear Differential Equation

A homogeneous linear differential equation is one where all terms are dependent on the unknown function and its derivatives, but there are no free-standing terms (constants or functions of independent variables). The general form is:
\[a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \ldots + a_1(x)y' + a_0(x)y = 0\]
In our specific case, we're interested in second-order equations such as:
\[y'' + p(x)y' + q(x)y = 0\]
Homogeneous equations often arise in physics and engineering because they model systems where the output is directly proportional to the input. They imply a linear relationship and their solutions, or response, change in a predictable way.
One key characteristic of these equations is that if two functions are solutions, then their linear combination is also a solution. This superposition principle is at the heart of solving these equations and is one of the reasons why their theory is rich and widely applicable.