# Solve a linear system of differential equations

I have yet another question, I am trying to solve a differential equation by transforming it into 2 linear differential equations and then get a solution. Everything goes smoothly, until the last bit where I obtain solutions. I have an equation: $$y^{(3)}-7y''+15y'-9y=0$$ Later I transform it into a matrix:\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 9 & -15 & 7 \\ \end{matrix} calculate eigen values, I also calculate the eigen vector matrix which is:\begin{matrix} 1 & 1 & 1 \\ 3 & 1 & 1 \\ 9 & -3 & 1 \\ \end{matrix} and use a formula $$J=P^{-1}MP$$ to obtain my matrix $$\begin{matrix} 3 & -2 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \\ \end{matrix}$$ I then transoform it into a linear system, and that's where I do not know how to solve: $$\left\{ \begin{array}{c} v_1'=3v_1-2v_2 \\ v_2'=3v_2 \\ v_3'=v_3 \end{array} \right.$$

Thank you very much in advance!

## 2 Answers

A triangular system of linear ODEs with constant coefficients is just slightly harder to solve than a diagonal one. Assuming you did the calculations up to this point correctly, it proceeds this way.

First you solve for $v_3$:

$$v_3' = v_3 \Rightarrow v_3 = c_3 e^t.$$

Then you solve for $v_2$:

$$v_2' = 3v_2 \Rightarrow v_2 = c_2 e^{3t}.$$

Now you can plug that into the equation for $v_1$:

$$v_1' = 3v_1 - 2v_2 = 3v_1 - 2 c_2 e^{3t}.$$

This is now an inhomogeneous linear scalar equation, which can be handled for example by the method of integrating factors, or by undetermined coefficients.

• Thank you sir! This answer explained a lot, I know my question was a bit dumb, and thank you for answering! And regarding the last question, I know how to take an exponential of a Jordan block, just learned it yesterday. But I think a linear system of differential equations can only be solved by exponential if X(0) is given, and in my case it's not. – user298093 Dec 13 '15 at 18:39
• @SuleymanOrazgulyyev You can still give the general solution of $x'=Ax$ as $e^{At} x_0$ where $x_0$ is a parameter. – Ian Dec 13 '15 at 18:42
• You are right, I did not think of that :)thank you, and have a nice day :) – user298093 Dec 13 '15 at 18:43
• Sir, regarding your answer I understood everything except the $$v_1$$, you have stopped at $$3v_1-2c_2e^{3t}$$, I believe we need to find $$v_1$$ as well. Can you please continue. – user298093 Dec 13 '15 at 19:16
• @SuleymanOrazgulyyev Are you familiar with the method of integrating factors for first order linear scalar equations? – Ian Dec 13 '15 at 19:18

The theory of linear differential equations is well developed. Here is the more general setting.

Consider an ODE systerm $$\dot{x}=Ax,\quad x(0)=x_0.$$

Then $$x(t)=P\textrm{diag}[e^{B_jt}]P^{-1}x_0$$ where $$P^{-1}AP=\textrm{diag}[B_1,\cdots,B_r]$$ is the Jordan for of $A$.

See for instance Chapter 1.8 in Differential Equations and Dynamical Systems by Perko and Lawrence.

• Thank you for you time sir. I am familiar with the formula that you gave, but in this example I'm only stuck with differential equations, and don't know how to get solutions for it. If you can please show me how to solve this I will instantly catch it, I have worked with DEs before, but not with linear versions. – user298093 Dec 13 '15 at 18:15
• The problem with this explanation is that you need to know how to take the exponential of a Jordan block. While it is quite obvious how to take the exponential of a $1 \times 1$ Jordan block, it is not as obvious how to deal with a $n \times n$ Jordan block. – Ian Dec 13 '15 at 18:31