Consider the homogeneous linear first-order system differential equations[br][center][i]x[/i]'=[i]ax[/i]+[i]by[/i][br][i]y[/i]'=[i]cx[/i]+[i]dy[/i][/center]which can be written in matrix form as [b][i]X[/i][/b]'=[b][i]AX,[/i][/b] where [b][i]A[/i][/b] is the coefficients matrix.[br][br]The following worksheet is designed to analyse the nature of the critical point (when [math]\Delta\ne0[/math]) and solutions of the linear system [b][i]X[/i][/b]'=[b][i]AX.[/i][/b][br][br][b]Notation:[/b][br][list][*]Determinant of [b]A[/b]: [math]\Delta[/math][/*][*]Trace of [i][b]A[/b][/i]: [math]\tau[/math][/*][*]Eigenvalues: [math]\lambda_1[/math], [math]\lambda_2[/math][br][/*][*]Eigenvectors: [math]v_1[/math], [math]v_2[/math][br][br][/*][/list][b]Note:[/b] The eigenvectors on the left-side screen are normalised. [br][br][b]Warning:[/b] The online version does not show the case when there is only one eigenvector. You need to download the file.