Differential Equations: The Wronskian and Reduction of Order

Given a differential equation, find the two linearly independent solutions and show that they form a fundamental set of solutions using the Wronskian.

Using the Wronskian method, determine if the functions $y_1 = 2t - 1$, $y_2 = t^2 + 5$, and $y_3 = 4t - 7$ are linearly independent.

For two functions $y_1 = e^{2x}$ and $y_2 = e^{-2x}$, determine if they are linearly independent using the Wronskian test.

Using the Wronskian test, determine if the functions $y_1 = \, \cos(x)$ and $y_2 = \, \sin(x)$ are linearly independent.

Given $t²$ is a solution to $t^2y'' + 3ty' - 8y = 0$ Solve using the reduction of orders method.

Use the reduction of order formula to find another solution to this differential equation: $9y'' - 12y' + 4y = 0$.