Mathematics II20805 marksNumerical: linear independence of solutionsApplications to difference equations
Show that the solutions of yk+2 - 4yk+1 + 3yk = 0 are linearly independent.
Show that the solutions of $y_{k+2} - 4y_{k+1} + 3y_k = 0$ are linearly independent. [5]
Homogeneous linear difference equation: $$y{k+2} - 4y{k+1} + 3yk = 0$$ Task: show the two fundamental solutions are linearly independent. Assume a solution of the form $yk = m^k$. Substituting: $$m^{k+2} - 4m^{k+1} + 3m^k = 0$$ Dividing by $m^k$ (with $m \neq 0$): $$m^2 - 4m + 3 = 0$$ $$(m-1)(m-3...