# Solving a first order ordinary differential equation

we have : $x'(t) = (x'_1(t), x'_2(t), x'_3(t))$ and $x(t)=(x_1(t), x_2(t), x_3(t))$ Then the system $x'(t)=Ax(t)+bu$ where A= \begin{matrix} -1 & 1 & 0 \\ 1 & -3 & 1 \\ 0 & 1 & -1 \\ \end{matrix} and b= \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} How can I solve the first order ODE :

$x'_1 = x_1 -3x_2 + x_3$ ?

• what are $x_2$ and $x_3$? – Dr. Sonnhard Graubner Oct 13 '15 at 15:05
• I have edited the question @Dr.SonnhardGraubner – user189013 Oct 13 '15 at 15:10
• You can solve for $x_1, x_2$ and $x_3$ given $x'(t) = Ax(t)+bu$, which is an inhomogeneous DE. Your question is to solve a homogeneous DE where it is not clear what $x_1, x_2$ and $x_3$ represent. Are they $x_1, x_2$ and $x_3$ from solving $x'(t) = Ax(t)+bu$? Is that really $x'_1$ on the left hand side? – Paul Oct 13 '15 at 15:45