Understanding the Characteristic Equation with Repeated Real Roots

📚 Understanding Characteristic Equations with Repeated Real Roots

In the realm of differential equations, the characteristic equation plays a pivotal role in finding solutions. When dealing with linear homogeneous differential equations with constant coefficients, repeated real roots in the characteristic equation require special attention. This guide provides a comprehensive overview.

📜 History and Background

The method of using characteristic equations to solve differential equations dates back to the 18th century, with significant contributions from mathematicians like Leonhard Euler and Jean-Baptiste le Rond d'Alembert. They recognized that exponential functions could satisfy certain differential equations, leading to the development of the characteristic equation as a means to find these solutions.

🔑 Key Principles

• 🔢 Definition: The characteristic equation is an algebraic equation derived from a linear homogeneous differential equation. For a second-order equation of the form $ay'' + by' + cy = 0$, the characteristic equation is $ar^2 + br + c = 0$.
• 🌱 Repeated Roots: When the discriminant ($b^2 - 4ac$) is zero, the characteristic equation has a repeated real root, denoted as $r$.
• 📝 General Solution: If $r$ is a repeated real root, the general solution to the differential equation is given by $y(x) = c_1e^{rx} + c_2xe^{rx}$, where $c_1$ and $c_2$ are arbitrary constants.
• 💡 Why $xe^{rx}$?: The term $xe^{rx}$ arises because $e^{rx}$ is already a solution. Multiplying by $x$ provides a linearly independent solution, ensuring the general solution spans the solution space.

🧮 Finding the General Solution: A Step-by-Step Approach

1. Step 1: Form the Characteristic Equation: Replace $y''$ with $r^2$, $y'$ with $r$, and $y$ with 1 in the given differential equation.
2. Step 2: Solve for the Roots: Solve the quadratic equation $ar^2 + br + c = 0$. If the discriminant is zero, you have repeated real roots.
3. Step 3: Write the General Solution: If $r$ is the repeated root, the general solution is $y(x) = c_1e^{rx} + c_2xe^{rx}$.

➗ Real-World Examples

Example 1:

Consider the differential equation $y'' - 4y' + 4y = 0$.

1. The characteristic equation is $r^2 - 4r + 4 = 0$.
2. Solving for $r$, we get $(r - 2)^2 = 0$, so $r = 2$ is a repeated real root.
3. The general solution is $y(x) = c_1e^{2x} + c_2xe^{2x}$.

Example 2:

Consider the differential equation $y'' + 6y' + 9y = 0$.

1. The characteristic equation is $r^2 + 6r + 9 = 0$.
2. Solving for $r$, we get $(r + 3)^2 = 0$, so $r = -3$ is a repeated real root.
3. The general solution is $y(x) = c_1e^{-3x} + c_2xe^{-3x}$.

📝 Practice Quiz

Solve the following differential equations:

1. $y'' - 2y' + y = 0$
2. $4y'' + 4y' + y = 0$

📈 Conclusion

Understanding characteristic equations with repeated real roots is crucial for solving linear homogeneous differential equations. By correctly identifying repeated roots and applying the appropriate form of the general solution, you can effectively solve a wide range of problems. This knowledge is fundamental in various fields, including physics, engineering, and applied mathematics.