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

[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.

 

[pasmith]

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 [itex]e^{6t}[/itex] come from? [itex]e^{2t}[/itex] is an eigenfunction of the derivative operator: [itex](e^{2t})' = 2e^{2t}[/itex]. So I would expect the particular function to be [itex]ae^{2t}[/itex] for some constant vector [itex]a[/itex], 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 [itex]e^{6t}[/itex] 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

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.