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Q1
Answer
For the differential equations $\dfrac{\mathrm{d}x}{\mathrm{d}t} = 3x - y$ and $\dfrac{\mathrm{d}y}{\mathrm{d}t} = x + y$.
  1. Find the general solutions for $x$ and $y$.
    1. Given that $x = 3$ and $y = 1$ when $t = 0$, find the particular solutions.
    1. $x = (A + Bt)e^{2t}$, $\;y = ((A - B) + Bt)e^{2t}$
    2. $x = (3 + t)e^{2t}$, $\;y = (2 + t)e^{2t}$
    Q2
    Answer
    For the differential equations $\dfrac{\mathrm{d}x}{\mathrm{d}t} = 4x - y$ and $\dfrac{\mathrm{d}y}{\mathrm{d}t} = 2x + y$.
    1. Find the general solutions for $x$ and $y$.
      1. Given that $x = 2$ and $y = 3$ when $t = 0$, find the particular solutions.
      2. Describe the behaviour of $x$ and $y$ as $t \to \infty$.
      1. $x = Ae^{2t} + Be^{3t}$, $\;y = 2Ae^{2t} + Be^{3t}$
      2. $x = Ae^{2t} + Be^{3t}$; apply $x(0) = 2$, $y(0) = 3$ to find $A = 1$, $B = 1$; $x = e^{2t} + e^{3t}$, $y = 2e^{2t} + e^{3t}$
      3. Both $x$ and $y$ tend to infinity
      Q3
      Answer
      Given that $x = 0$ and $y = \dfrac{1}{5}$ at $t = 0$, find the particular solutions to $\dfrac{\mathrm{d}x}{\mathrm{d}t} = 2x - 3y - 2$ and $\dfrac{\mathrm{d}y}{\mathrm{d}t} = x + y - 1$.
        $x = -\dfrac{2}{5}e^{4t} + e^{-t} - \dfrac{3}{5}$ and $y = \dfrac{4}{15}e^{4t} + e^{-t} - \dfrac{16}{15}$
        Q4
        Answer
        For the differential equations $\dfrac{\mathrm{d}x}{\mathrm{d}t} = -x + 4y$ and $\dfrac{\mathrm{d}y}{\mathrm{d}t} = -x - y + e^t$.
        1. Write down the general solutions for $x$ and $y$.
          1. Given $x = 0$ and $y = 1$ when $t = 0$, find the particular solutions.
          1. $x = e^{-t}(A\cos 2t + B\sin 2t) + \dfrac{1}{2}e^t$; $y = \dfrac{1}{4}e^t + \dfrac{1}{2}e^{-t}(B\cos2t - A\sin 2t)$
          2. $A = -\dfrac{1}{2}$, $B = \dfrac{3}{2}$
          Q5
          Answer
          Two populations of animals, prey $x$ and predator $y$ (in thousands), satisfy the linearised equations $\dfrac{\mathrm{d}x}{\mathrm{d}t} = 3x - 2y$ and $\dfrac{\mathrm{d}y}{\mathrm{d}t} = 2x - y$, where $t$ is time in years.
          1. Find the general solutions for $x$ and $y$.
            1. Given that $x = 10$ and $y = 8$ when $t = 0$, find the particular solutions.
            2. Find the long-term behaviour of both populations.
            1. $x = (A + Bt)e^{t}$, $\;y = \left(A - \dfrac{B}{2} + Bt\right)e^t$
            2. $A = 10$, $B = 4$; $\;x = (10 + 4t)e^t$, $\;y = (8 + 4t)e^t$
            3. Both populations tend to infinity
            Q6
            Answer
            For the differential equations $\dfrac{\mathrm{d}x}{\mathrm{d}t} = 2x + y + t$ and $\dfrac{\mathrm{d}y}{\mathrm{d}t} = x + 2y - t$.
            1. Find the general solutions for $x$ and $y$.
              1. Given that $x = 1$ and $y = 0$ when $t = 0$, find the particular solutions.
              1. $x = Ae^t + Be^{3t} - t - 1$, $\;y = -Ae^t + Be^{3t} + t + 1$
              2. $A = \dfrac{3}{2}$, $B = \dfrac{1}{2}$; $\;x = \dfrac{3}{2}e^t + \dfrac{1}{2}e^{3t} - t - 1$, $\;y = -\dfrac{3}{2}e^t + \dfrac{1}{2}e^{3t} + t + 1$
              Q7
              Answer
              Find the general solution of $\dfrac{\mathrm{d}y}{\mathrm{d}x} + 2y = 6$.
                $y = 3 + Ce^{-2x}$
                Q8
                Answer
                Find the general solution of $\dfrac{\mathrm{d}y}{\mathrm{d}x} - 3y = e^{5x}$.
                  $y = \dfrac{1}{2}e^{5x} + Ce^{3x}$
                  Q9
                  Answer
                  Find the general solution of $\dfrac{\mathrm{d}y}{\mathrm{d}x} + \dfrac{y}{x} = x^2$, for $x > 0$.
                    $y = \dfrac{x^3}{4} + \dfrac{C}{x}$
                    Q10
                    Answer
                    Solve $\dfrac{\mathrm{d}y}{\mathrm{d}x} + y\cot x = \cos x$, given that $y = 0$ when $x = \dfrac{\pi}{2}$.
                      $y = \dfrac{x\sin x - \cos x - \frac{\pi}{2} + 1}{\sin x}$
                      Q11
                      Answer
                      Find the general solution of $x\dfrac{\mathrm{d}y}{\mathrm{d}x} + 2y = \ln x$, for $x > 0$.
                        $y = \dfrac{\ln x}{3x^2} - \dfrac{1}{9x^2} + \dfrac{C}{x^2}$
                        Q12
                        Answer
                        Solve $\dfrac{\mathrm{d}y}{\mathrm{d}x} - \dfrac{2y}{x} = x^2e^x$, given that $y = 0$ when $x = 1$, for $x > 0$.
                          $y = x^2(e^x - e)$