Name: MAT 214- Diff. Eq. and Series Assi gnment: Section 4.3 Homogeneous Linear Equations With Constant...

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Name: MAT 214- Diff. Eq. and Series Assi gnment: Section 4.3 Homogeneous Linear Equations With Constant Coefficients. 1. Veri

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$\small \textbf{Consider the set}\\ S=\left \{ e^{k_1x} ,\: e^{k_2x} \right \}\\ \textbf{We now compute the Wronskian of two solutions}\\ W(e^{k_1x} ,\: e^{k_2x} )=\begin{vmatrix} e^{k_1x} &e^{k_2x} \\ k_1 e^{k_1x} &k_2e^{k_2x} \end{vmatrix}\\ =k_2e^{(k_1+k_2)x} -k_1e^{(k_1+k_2)x} \\ =e^{(k_1+k_2)x} (k_2-k_1)\\ W(e^{k_1x} ,\: e^{k_2x} )\neq 0,\quad (k_2\neq k_1)\: \: \text{for all x}\\ \textbf{Hence, the two solutions are called a fundamental set of solutions.}$

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