# Conversion of Nonhomogeneous differential equations system into nonhomogeneous 2nd order differential equation

I want to convert the following 1st order non homogeneous differential equation system into 2nd order differential equation.

$$\vec{x}'=\small\begin{pmatrix}1&2\\3&2\end{pmatrix}\vec{x}+t\small\begin{pmatrix}2\\-4\end{pmatrix}$$

So i get $$x'_1=x_1+2x_2+2t$$ and $$x'_2=3x_1+2x_2-4t$$

After doing some calculations, I get the following differential equation $$x''_1-3x'_1-4x_1-10t=0$$ Is this correct? I want to verify it by checking general and particular solution to this 2nd order differential equation. I already have general and particular solution to 1st order differential equation system

We have

$$x' = x + 2 y + 2 t \\ y' = 3 x + 2 y - 4t$$

From the first equation, we have

$$x'' = x' + 2 y' + 2 = x' + 2(3x + 2 y - 4t)+ 2 = x' + 2(3x + (x'-x-2t) - 4t)+2$$

Simplifying

$$x'' - 3x' - 4x + 12 t - 2 = 0$$

• +1 Op mixed 12t and 2... – Isham Nov 10 '18 at 19:28
• @Moo,If i want to convert this 2nd order differential equation into 1st order differential equation system, how to convert it. We got $x''-3x'-4x=-12t+2$ – Dhamnekar Winod Nov 11 '18 at 9:16
• @DhamnekarWinod: See tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx – Moo Nov 11 '18 at 12:27
• @Moo,I got $x'=3x+2x^2-6t^2+2t$. Is it correct? – Dhamnekar Winod Nov 16 '18 at 6:34
• No, that is not correct. $x$ depends on $t$ and you cannot do that. Try the process and examples at Undetermined Coefficients. You can also use things like Variation of Parameters or Laplace Transforms. I am surprised, hasn't your class and/or book reviewed these things? – Moo Nov 16 '18 at 12:08