Let [math]x_1,x_2[/math] and [math]x_3[/math] be functions of [math]t[/math] that satisfy the following system of ordinary differential equations (ODEs):[br][br][math]\left\{\begin{eqnarray}\dot{x}_1 & = & 4x_1+x_2-x_3\\ \dot{x}_2 & = & 2x_1+5x_2-2x_3\\ \dot{x}_3 & = & x_1+x_2+2x_3\end{eqnarray}\right.[/math][br][br]where [math]\dot{x}_i[/math] means [math]\frac{dx_i}{dt}[/math] for [math]i=1,2,3[/math]. The three variable functions are said to be [b]coupled[/b] to each other.[br][br]First of all, we rewrite the above system as a matrix equation:[br][br][math]\begin{pmatrix}\dot{x}_1\\ \dot{x}_2 \\ \dot{x}_3\end{pmatrix}=\begin{pmatrix}4&1&-1\\2&5&-2\\1&1&2\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}[/math], or more compactly, [math]\dot{x}=Ax[/math].[br][br]We diagonalize [math]A[/math] such that [math]A=PDP^{-1}[/math]:[br][br][math]\begin{pmatrix}4&1&-1\\2&5&-2\\1&1&2\end{pmatrix}=\begin{pmatrix}-1&1&1\\1&0&2\\0&1&1\end{pmatrix}\begin{pmatrix}3&0&0\\0&3&0\\0&0&5\end{pmatrix}\begin{pmatrix}-1&0&1\\-\frac12&-\frac12&\frac32\\ \frac12&\frac12&-\frac12\end{pmatrix}[/math][br][br]So [math]\dot{x}=PDP^{-1}x \Rightarrow P^{-1}\dot{x}=DP^{-1}x[/math][br][br]We make a change of variables: [math]y=P^{-1}x[/math]. It is easy to see that [math]\dot{y}=P^{-1}\dot{x}[/math]. Therefore, [math]\dot{y}=P^{-1}\dot{x}=DP^{-1}x=Dy[/math]:[br][br][math]\begin{pmatrix}\dot{y}_1\\ \dot{y}_2 \\ \dot{y}_3\end{pmatrix}=\begin{pmatrix}3&0&0\\0&3&0\\0&0&5\end{pmatrix}\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}=\begin{pmatrix}3y_1\\3y_2\\5y_3\end{pmatrix}[/math][br][br]This system of ODEs is very easy to solve as it is [b]decoupled[/b]:[br][br][math]\begin{pmatrix}y_1\\ y_2 \\ y_3\end{pmatrix}=\begin{pmatrix}c_1e^{3t}\\ c_2e^{3t} \\ c_3e^{5t}\end{pmatrix}[/math], where [math]c_1,c_2,c_3[/math] are constants.[br][br]Hence, we change the variables back to [math]x[/math]: [math]y=P^{-1}x\Rightarrow x=Py[/math] and we have[br][br][math]\begin{eqnarray}\begin{pmatrix}x_1\\ x_2 \\ x_3\end{pmatrix}&=&\begin{pmatrix}-1&1&1\\1&0&2\\0&1&1\end{pmatrix}\begin{pmatrix}y_1\\ y_2 \\ y_3\end{pmatrix}\\&=&\begin{pmatrix}-c_1e^{3t}+c_2e^{3t}+c_3e^{5t}\\c_1e^{3t}+2c_3e^{5t}\\c_2e^{3t}+c_3e^{5t}\end{pmatrix}\end{eqnarray}[/math][br][br]