SECTION B Answer ANY TWO (2) Questions from this section. 1. a) \( \begin{array}{l}\text { Show that the functions } y_{1}=e^{-3 x}, y_{2}=\cos 2 x \text { and } y_{3}=\sin 2 x \\ \text { form a fundamental set of solutions for a certain third order ordinary } \\ \text { differential equation. }\end{array} \) [8 marks]

To show that the functions \( y_1 = e^{-3x}, y_2 = \cos(2x), \) and \( y_3 = \sin(2x) \) form a fundamental set of solutions, one typically constructs the Wronskian determinant \( W(y_1, y_2, y_3) \). If \( W(y_1, y_2, y_3) \neq 0 \) at some point in the interval of interest, then the functions are linearly independent, thus forming a fundamental set of solutions for a third-order ODE. The Wronskian can be calculated as follows: \[ W(y_1, y_2, y_3) = \begin{vmatrix} e^{-3x} & \cos(2x) & \sin(2x) \\ -3e^{-3x} & -2\sin(2x) & 2\cos(2x) \\ 9e^{-3x} & -4\cos(2x) & -4\sin(2x) \end{vmatrix} \] Calculating this determinant will confirm their linear independence, demonstrating that they form a fundamental set for the corresponding third-order ODE. Moving forward, you can also verify the specific ODE they satisfy. By taking each function's derivatives and substituting them into an assumed third-order differential equation, you may derive that they indeed fulfill the required relationships to validate their fundamental nature. Always remember to check the linear independence through the Wronskian cleverly constructed and the calculated derivatives to ensure that each solution contributes uniquely to the solution space!