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Could someone teach me how to solve Question 1 & 3? I have read over the corresponding chapters in my textbook but I still cannot understand how to solve them. I would like to know the steps on how to solve these questions so that I will be able to re-take the questions (randomised so I want to practice the problems multiple times after I learn how to solve them.)

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Question 1. You have system $$ \frac{d X}{dt} = X + 5 Y,\\ \frac{d Y}{dt} = 2 X + 3 Y. $$ Substituting $Y$ from the first equation to the second one, we ge $$ \frac{d Y}{dt} = 2 X + \frac{3}{5} \frac{d X}{dt} - \frac{3}{5} X = \frac{7}{5} X + \frac{3}{5} \frac{d X}{dt}. $$

At the same time, if we differentiate the first equation, then we gen $$ \frac{d^2 X}{d t^2} = \frac{d X}{dt} + 5 \frac{d Y}{dt}. $$

Now, we substitute here $\frac{d Y}{dt}$: $$ \frac{d^2 X}{d t^2} = \frac{d X}{dt} + 7 X + 3\frac{d X}{dt} = 4 \frac{d X}{dt} + 7X. $$

Hence, $$ \frac{d^2 X}{d t^2} - 4 \frac{d X}{dt} - 7X = 0, $$ and so $a = -4$ and $b = -7$.

Question 2. Just substitute the solutions to the differential equations and initial conditions. Then, you will get the standard system of 4 linear equations for 4 unknown coefficients.

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