Understanding linearly dependent vs independent solutions in DEs: A comprehensive guide

📚 Understanding Linearly Dependent vs. Independent Solutions in Differential Equations

Okay, let's break down the difference between linearly dependent and linearly independent solutions in differential equations. It's a crucial concept for finding the general solution!

Definition of Linearly Dependent Solutions: Solutions $y_1, y_2, ..., y_n$ are linearly dependent if there exist constants $c_1, c_2, ..., c_n$, not all zero, such that:

$c_1y_1 + c_2y_2 + ... + c_ny_n = 0$

This means one solution can be expressed as a linear combination of the others.

Definition of Linearly Independent Solutions: Solutions $y_1, y_2, ..., y_n$ are linearly independent if the only constants $c_1, c_2, ..., c_n$ that satisfy:

$c_1y_1 + c_2y_2 + ... + c_ny_n = 0$

are $c_1 = c_2 = ... = c_n = 0$. In other words, no solution can be written as a linear combination of the others.

📝 Side-by-Side Comparison

🔑 Key Takeaways

• ➕ Linear Dependence: Think of it as redundancy. One solution is essentially a scaled version (or combination) of another.
• ➖ Linear Independence: Each solution contributes unique information to the overall solution.
• ➗ Wronskian: The Wronskian is a determinant that helps determine linear independence. If it's zero, the solutions are likely linearly dependent. If it's non-zero, they are linearly independent.
• 💡 General Solution: To form the general solution of an $n$-th order linear homogeneous differential equation, you need $n$ linearly independent solutions.
• 📈 Example: Consider the differential equation $y'' + y = 0$. Two solutions are $y_1 = \sin(x)$ and $y_2 = \cos(x)$. These are linearly independent. The general solution is $y = c_1\sin(x) + c_2\cos(x)$.
• ✍️ Wronskian Calculation: For two functions $f(x)$ and $g(x)$, the Wronskian is: $W(f, g) = \begin{vmatrix} f & g \\ f' & g' \end{vmatrix} = fg' - gf'$.
• ✅ Checking for Linear Independence: If $W(f, g) \neq 0$, then $f$ and $g$ are linearly independent.