91Ó°ÊÓ

A fundamental set of solutions for a differential equation consists of solutions that can be combined to express the general solution for a given differential equation. For second-order linear homogeneous differential equations like \( y'' + 8y' + 16y = 0 \), the general solution is a linear combination of two independent solutions. In our example:

• We have \( y_1(t) = e^{-4t} \) and \( y_2(t) = te^{-4t} \) as candidate solutions.
• Both solutions satisfy the differential equation, verified by substituting them back into the original equation and confirming they equal zero.

They form the fundamental set because they are independent (not multiples of each other), so any solution to the differential equation can be written as \( y(t) = c_1 y_1 + c_2 y_2 \), where \( c_1 \) and \( c_2 \) are constants determined by initial conditions.

The Wronskian is a determinant used to test whether two solutions to a differential equation are linearly independent. For functions \( y_1 \) and \( y_2 \), the Wronskian is given by:\[W(y_1, y_2) = \begin{vmatrix}y_1 & y_2 \y_1' & y_2'\end{vmatrix}\]For our solutions, \( y_1(t) = e^{-4t} \) and \( y_2(t) = te^{-4t} \), their Wronskian is calculated as:\[W(y_1, y_2) = e^{-4t}(e^{-4t} - 4te^{-4t}) + 4te^{-8t}\]\[= e^{-8t}\]Observing that the Wronskian is not identically zero, \( y_1 \) and \( y_2 \) are linearly independent. This confirms they form a fundamental set of solutions.

An initial value problem consists of a differential equation along with conditions specified at a particular point. Solving it involves finding a specific solution that meets these conditions. Our problem provides the differential equation:\[ y'' + 8y' + 16y = 0 \]and initial conditions:

• \( y(0) = 2 \)
• \( y'(0) = -1 \)

Using the fundamental set of solutions \( y(t) = c_1 e^{-4t} + c_2 te^{-4t} \), we can apply the initial conditions:

• \( y(0) = 2 \) gives \( c_1 = 2 \)
• \( y'(0) = -1 \) helps us solve \( -8 + c_2 = -1 \), thus \( c_2 = 7 \)

This results in a particular solution that satisfies both the differential equation and the initial conditions.

Linear independence is a critical concept in determining whether solutions to a differential equation can form a fundamental set. For two functions \( y_1 \) and \( y_2 \), they are linearly independent if the equation:\[ c_1 y_1(t) + c_2 y_2(t) = 0 \]for all \( t \) implies \( c_1 = 0 \) and \( c_2 = 0 \). If this is the case, neither function is a scalar multiple of the other.The Wronskian test provides an efficient method to check for independence. Since the Wronskian \( W(y_1, y_2) = e^{-8t} \) in our example doesn’t equate to zero for all \( t \), \( y_1 \) and \( y_2 \) are linearly independent.Linear independence ensures the solutions can span the solution space of the differential equation, allowing us to construct any solution as a linear combination of \( y_1 \) and \( y_2 \).