How to check a particular solution of System of Linear ODEs?

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Hall
If I have been given a system of inhomogeneous linear ODEs,
$$\vec{x'} = \begin{bmatrix} 4 & -1 \\ 5 & -2 \\ \end{bmatrix} \vec{x} + \begin{bmatrix} 18e^{2t} \\ 30e^{2t}\\ \end{bmatrix}$$

I have found its particular solution to be:
$$1/4 \begin{bmatrix} -31e^{2t} - 25e^{6t} \\ 85e^{2t} - 25e^{6t} \\ \end{bmatrix}$$

But this answer doesn't match with the answer given in the book. Can someone tell me how to check if this solutions works by writing some code in Mathematica? I know, I can use DSolve for solving them, but I'm asking a reverse of that.

Please guide me step by step, I'm new to Mathematica and I don't have any background in programming.

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Homework Helper
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If I have been given a system of inhomogeneous linear ODEs,
$$\vec{x'} = \begin{bmatrix} 4 & -1 \\ 5 & -2 \\ \end{bmatrix} \vec{x} + \begin{bmatrix} 18e^{2t} \\ 30e^{2t}\\ \end{bmatrix}$$

I have found its particular solution to be:
$$1/4 \begin{bmatrix} -31e^{2t} - 25e^{6t} \\ 85e^{2t} - 25e^{6t} \\ \end{bmatrix}$$

But this answer doesn't match with the answer given in the book.

I look at your solution, and I must ask myself: Where does $e^{6t}$ come from? $e^{2t}$ is an eigenfunction of the derivative operator: $(e^{2t})' = 2e^{2t}$. So I would expect the particular function to be $ae^{2t}$ for some constant vector $a$, which can be determined by substituting this into the ODE. That your answer is not of this form, or of this form plus a complementary function (6 is not an eigenvalue of the matrix, so a multiple of $e^{6t}$ is not a complementary function), leads me to suspect that you have made an error, but since you haven't shown your working I can't tell you what it is.

Can someone tell me how to check if this solutions works by writing some code in Mathematica? I know, I can use DSolve for solving them, but I'm asking a reverse of that.

Please guide me step by step, I'm new to Mathematica and I don't have any background in programming.

The answer to this question is "symbolic differentiation". Define your proposed solution as a function, and check to see that
Code:
f'[t] - {{4, -1},{5, -2}} . f[t] - {18*exp[2*t], 30*exp[2*t]}
is zero. Have a look at the examples at https://reference.wolfram.com/language/ref/Derivative.html.

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Hall