# differences in the solutions of DE system and its converted 2nd order Differential equation

#### Dhamnekar Winod

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

Now i got the solution to this differential equation system as

$\vec{x}(t)=c_1e^{-t}\small\begin{pmatrix}-1\\1\end{pmatrix}$+$c_2e^{4t}\small\begin{pmatrix}2\\3\end{pmatrix}$+$t\small\begin{pmatrix}3\\\frac{-5}{2}\end{pmatrix}$+$\small\begin{pmatrix}-2.75\\2.875\end{pmatrix}$

Now i converted this differential equation system into ordinary differential equation $y''-3y'-4y+12t-2=0$

I got solution to this DE as $y=C_1e^{-x}+C_2e^{4x}+3t-\frac12$.

Now my question why there is diference in these two solutions?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hello,
$\vec{x'}=\small\begin{pmatrix}1&2\\3&2\end{pmatrix}\vec{x}+t\small\begin{pmatrix}2\\-4\end{pmatrix}$

Now i got the solution to this differential equation system as

$\vec{x}(t)=c_1e^{-t}\small\begin{pmatrix}-1\\1\end{pmatrix}$+$c_2e^{4t}\small\begin{pmatrix}2\\3\end{pmatrix}$+$t\small\begin{pmatrix}3\\\frac{-5}{2}\end{pmatrix}$+$\small\begin{pmatrix}-2.75\\2.875\end{pmatrix}$

Now i converted this differential equation system into ordinary differential equation $y''-3y'-4y+12t-2=0$

I got solution to this DE as $y=C_1e^{-x}+C_2e^{4x}+3t-\frac12$.

Now my question why there is diference in these two solutions?
Hi Dhamnekar Winod, welcome to MHB!

Your solution contains $x$ on the right hand side. I presume it should be $t$?
Either way, if I feed the latter equation to Wolfram|Alpha, I get:
$$y(t)=c_1e^{-t}+c_2e^{4t}+3t-\frac{11}{4}$$
So:
• It seems to be the solution for $x(t)$ rather than $y(t)$.
• Your equation is correct, although for $x(t)$ rather than $y(t)$.
• There is a minor arithmetic mistake somewhere in your solution.

#### Dhamnekar Winod

##### Active member
Hello, The derivative of $x_1=C_1e^{-t}+C_23e^{4t}-2.5t+2.875$in the differential equation system.But in the second case,it is different Why?