Steps to construct the general solution from a fundamental set of solutions

📚 Definition of Fundamental Solutions and General Solutions

In the realm of linear homogeneous differential equations, a fundamental set of solutions refers to a set of linearly independent solutions that span the solution space. In simpler terms, you can create any other solution to the equation by combining these fundamental solutions. The general solution is then the expression that captures all possible solutions, formed by a linear combination of the fundamental set.

📜 Historical Context

The development of methods for solving differential equations dates back to the 17th century, with contributions from giants like Newton and Leibniz. The concept of a fundamental set of solutions and the general solution emerged as mathematicians sought to understand the complete behavior of these equations. The theory was further refined in the 18th and 19th centuries, solidifying its place in mathematical analysis.

🔑 Key Principles

• 🔢 Linear Homogeneity: The differential equation must be linear and homogeneous. A linear equation means that if $y_1$ and $y_2$ are solutions, then $c_1y_1 + c_2y_2$ is also a solution for any constants $c_1$ and $c_2$. Homogeneous means that the equation is equal to zero.
• 🌱 Linear Independence: The solutions in the fundamental set must be linearly independent. This means that no solution can be written as a linear combination of the others. Formally, if $c_1y_1 + c_2y_2 + ... + c_ny_n = 0$ for all $x$, then $c_1 = c_2 = ... = c_n = 0$.
• 📐 Spanning the Solution Space: The fundamental set must span the entire solution space. This means that any solution to the differential equation can be expressed as a linear combination of the fundamental set.
• ➕ General Solution Formula: If $y_1, y_2, ..., y_n$ form a fundamental set of solutions, then the general solution $y(x)$ is given by: $y(x) = c_1y_1(x) + c_2y_2(x) + ... + c_ny_n(x)$, where $c_1, c_2, ..., c_n$ are arbitrary constants.

✍️ Steps to Construct the General Solution

• 🔎 Find a Fundamental Set: Obtain $n$ linearly independent solutions $y_1(x), y_2(x), ..., y_n(x)$ to the $n$-th order linear homogeneous differential equation. This can involve techniques like finding characteristic roots or using reduction of order.
• ➕ Form a Linear Combination: Create a linear combination of these fundamental solutions, each multiplied by an arbitrary constant: $y(x) = c_1y_1(x) + c_2y_2(x) + ... + c_ny_n(x)$
• ✅ Verify Linear Independence: Ensure that the fundamental solutions are linearly independent. This can be checked using the Wronskian.
• 🎯 State the General Solution: The linear combination is the general solution. It represents all possible solutions to the differential equation.

🧪 Real-World Examples

Example 1: Second-Order Linear Homogeneous Equation

Consider the differential equation $y'' - 3y' + 2y = 0$.

1. The characteristic equation is $r^2 - 3r + 2 = 0$, which factors as $(r-1)(r-2) = 0$. The roots are $r_1 = 1$ and $r_2 = 2$.
2. Thus, the fundamental set of solutions is {$e^x$, $e^{2x}$}.
3. The general solution is $y(x) = c_1e^x + c_2e^{2x}$.

Example 2: Third-Order Linear Homogeneous Equation

Consider the differential equation $y''' - 6y'' + 11y' - 6y = 0$.

1. The characteristic equation is $r^3 - 6r^2 + 11r - 6 = 0$, which factors as $(r-1)(r-2)(r-3) = 0$. The roots are $r_1 = 1$, $r_2 = 2$, and $r_3 = 3$.
2. Thus, the fundamental set of solutions is {$e^x$, $e^{2x}$, $e^{3x}$}.
3. The general solution is $y(x) = c_1e^x + c_2e^{2x} + c_3e^{3x}$.

💡 Conclusion

Understanding how to construct the general solution from a fundamental set of solutions is crucial for solving linear homogeneous differential equations. By finding a set of linearly independent solutions and forming a linear combination, we can express all possible solutions to the equation. This is a foundational concept in differential equations and has numerous applications in physics, engineering, and other fields.