Steps to prove the Sturm Separation Theorem for second-order ODEs.

📚 What is the Sturm Separation Theorem?

The Sturm Separation Theorem is a powerful result in the study of second-order linear homogeneous ordinary differential equations (ODEs). It tells us about the oscillatory behavior of solutions, specifically how the zeros (or roots) of linearly independent solutions are intertwined.

📜 Historical Background

The theorem is named after Jacques Charles François Sturm, a 19th-century mathematician who made significant contributions to various areas of mathematics, including differential equations. His work laid the foundation for understanding the qualitative behavior of solutions to ODEs.

🔑 Key Principles and Assumptions

The Sturm Separation Theorem applies to second-order linear homogeneous ODEs of the form:

$- (p(x)y')' + q(x)y = 0$,

or equivalently,

$p(x)y'' + p'(x)y' + q(x)y = 0$,

where $p(x) > 0$ and $p(x)$, $p'(x)$, and $q(x)$ are continuous on the interval of interest.

📝 Steps to Prove the Sturm Separation Theorem

Here's a breakdown of the steps involved in proving the theorem:

1️⃣ 🚀 Start with the ODE:

• 🔍 Consider the second-order linear homogeneous ODE: $p(x)y'' + p'(x)y' + q(x)y = 0$
• 📖 Assume $p(x) > 0$ and that $p(x)$, $p'(x)$, and $q(x)$ are continuous on the interval $[a, b]$.

2️⃣ 🧬 Define Linearly Independent Solutions:

• 🔬 Let $y_1(x)$ and $y_2(x)$ be two linearly independent solutions of the given ODE.
• 🧪 This means that no constant $c$ exists such that $y_1(x) = c y_2(x)$ for all $x$ in $[a, b]$.

3️⃣ 🔢 Assume $y_1(x)$ has Consecutive Zeros:

• 📊 Suppose $y_1(x)$ has two consecutive zeros at $x_1$ and $x_2$ in the interval $[a, b]$, where $x_1 < x_2$. This means $y_1(x_1) = 0$ and $y_1(x_2) = 0$.

4️⃣ 🌍 Show $y_2(x)$ has a Zero between $x_1$ and $x_2$:

• 💡 We want to prove that $y_2(x)$ must have at least one zero in the open interval $(x_1, x_2)$. We proceed by contradiction.
• 🚫 Assume $y_2(x)$ has no zeros in $(x_1, x_2)$. This means $y_2(x) > 0$ or $y_2(x) < 0$ for all $x$ in $(x_1, x_2)$. Without loss of generality, assume $y_2(x) > 0$ in $(x_1, x_2)$.

5️⃣ ✍️ Use Abel's Identity (Wronskian):

• 📝 The Wronskian of two solutions $y_1(x)$ and $y_2(x)$ is defined as $W(x) = y_1(x)y_2'(x) - y_1'(x)y_2(x)$.
• 🔑 Abel's Identity states that $W(x) = C/p(x)$ for some constant $C$, where $p(x)$ is the coefficient from the ODE. Since $y_1$ and $y_2$ are linearly independent, $C \neq 0$.

6️⃣ 📈 Evaluate the Wronskian at $x_1$ and $x_2$:

• 📌 At $x_1$, $W(x_1) = y_1(x_1)y_2'(x_1) - y_1'(x_1)y_2(x_1) = -y_1'(x_1)y_2(x_1)$.
• 📍 At $x_2$, $W(x_2) = y_1(x_2)y_2'(x_2) - y_1'(x_2)y_2(x_2) = -y_1'(x_2)y_2(x_2)$.

7️⃣ 🧮 Analyze the Signs:

• ✅ Since $y_2(x) > 0$ in $(x_1, x_2)$, $y_2(x_1) \geq 0$ and $y_2(x_2) \geq 0$. Also, since $y_1(x_1) = y_1(x_2) = 0$ and $y_1(x)$ changes sign at $x_1$ and $x_2$, $y_1'(x_1)$ and $y_1'(x_2)$ have opposite signs.
• ➗ Therefore, $W(x_1)$ and $W(x_2)$ have the same sign.
• ➖ This implies that $C/p(x_1)$ and $C/p(x_2)$ have the same sign. Since $p(x) > 0$, $C$ must have the same sign at both points.

8️⃣ ❗ Derive the Contradiction:

• 🔥 However, since $W(x) = C/p(x)$, and $C$ is a nonzero constant, $W(x)$ has a constant sign. But the Wronskian also equals $-y_1'(x)y_2(x)$ evaluated at $x_1$ and $x_2$. Since $y_1'(x_1)$ and $y_1'(x_2)$ have opposite signs, the Wronskian *must* change sign between $x_1$ and $x_2$, which is a contradiction!

9️⃣ 🎉 Conclude the Proof:

• 🎈 Therefore, our assumption that $y_2(x)$ has no zeros in $(x_1, x_2)$ must be false. Hence, $y_2(x)$ must have at least one zero in the interval $(x_1, x_2)$. This completes the proof of the Sturm Separation Theorem.

💡 Real-World Examples

The Sturm Separation Theorem is vital in the analysis of physical systems described by ODEs, such as:

• 🎻 Vibration analysis: Understanding the modes of vibration of a string.
• ⚛️ Quantum mechanics: Analyzing the behavior of wave functions in quantum systems.
• 🌊 Wave propagation: Studying the propagation of waves in various media.

⭐ Conclusion

The Sturm Separation Theorem provides a fundamental understanding of the oscillatory behavior of solutions to second-order linear homogeneous ODEs. It showcases the intertwined nature of the zeros of linearly independent solutions and has broad applications in physics and engineering.