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Q1
Answer
Find the general solution of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} - 5\dfrac{\mathrm{d}y}{\mathrm{d}x} + 6y = 0$.
    $y = Ae^{2x} + Be^{3x}$
    Q2
    Answer
    Find the general solution of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 6\dfrac{\mathrm{d}y}{\mathrm{d}x} + 9y = 0$.
      $y = (A + Bx)e^{-3x}$
      Q3
      Answer
      Find the general solution of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} - 4\dfrac{\mathrm{d}y}{\mathrm{d}x} + 13y = 0$.
        $y = e^{2x}(A\cos 3x + B\sin 3x)$
        Q4
        Answer
        Find the general solution of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} - 3\dfrac{\mathrm{d}y}{\mathrm{d}x} - 10y = 2e^{4x}$.
          $y = Ae^{5x} + Be^{-2x} - \dfrac{1}{3}e^{4x}$
          Q5
          Answer
          Find the general solution of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 4y = 3\sin 2x$.
            $y = A\cos 2x + B\sin 2x - \dfrac{3}{4}x\cos 2x$
            Q6
            Answer
            Find the general solution of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 2\dfrac{\mathrm{d}y}{\mathrm{d}x} + 5y = 10x + 6$.
              $y = e^{-x}(A\cos 2x + B\sin 2x) + 2x + \dfrac{2}{5}$
              Q7
              Answer
              Solve $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} - 2\dfrac{\mathrm{d}y}{\mathrm{d}x} - 3y = 6$ given that $y = 0$ and $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0$ when $x = 0$.
                $y = \dfrac{1}{2}e^{3x} + \dfrac{3}{2}e^{-x} - 2$
                Q8
                Answer
                Solve $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 4\dfrac{\mathrm{d}y}{\mathrm{d}x} + 4y = 8e^{-2x}$ given that $y = 1$ and $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0$ when $x = 0$.
                  $y = (1 + 2x)e^{-2x} + 4x^2e^{-2x}$
                  Q9
                  Answer
                  Find the general solution of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} - y = xe^x$.
                    $y = Ae^x + Be^{-x} + \dfrac{1}{4}e^x(x^2 - x)$
                    Q10
                    Answer
                    A particle's displacement $x$ satisfies $\ddot{x} + 6\dot{x} + 8x = 0$, with $x = 4$ and $\dot{x} = -2$ when $t = 0$.
                    1. Find the particular solution satisfying the given initial conditions.
                      1. State, with a reason, whether the motion is under-damped, critically damped, or over-damped.
                        1. Determine whether the particle ever returns to the origin.
                          1. $x = 10e^{-2t} - 6e^{-4t}$
                          2. Over-damped: auxiliary equation has distinct real roots
                          3. Set $x = 0$: $e^{2t} = \dfrac{3}{5}$ has no positive solution — the particle never returns to the origin
                          Q11
                          Answer
                          The displacement $x$ of a damped oscillator satisfies $\ddot{x} + 2\dot{x} + 10x = 0$, with $x = 0$ and $\dot{x} = 6$ when $t = 0$.
                          1. Find the general solution.
                            1. Find the particular solution.
                              1. Find the first time at which $x = 0$ for $t > 0$.
                                1. Find the maximum displacement of the oscillator.
                                  1. $x = e^{-t}(A\cos 3t + B\sin 3t)$
                                  2. $x = 2e^{-t}\sin 3t$
                                  3. $t = \dfrac{\pi}{3}$
                                  4. $t = \dfrac{1}{3}\arctan 3 \approx 0.384$ and $x \approx 1.08$
                                  Q12
                                  Answer
                                  Find the general solution of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 9y = 4\cos 3x + 2\sin x$.
                                  1. Find the complementary function.
                                    1. Find a particular integral for $4\cos 3x$ only.
                                      1. Find a particular integral for $2\sin x$ only.
                                        1. Hence write down the general solution.
                                          1. $A\cos 3x + B\sin 3x$
                                          2. $y\dfrac{2}{3}x\sin 3x$
                                          3. $\dfrac{1}{4}\sin x$
                                          4. $y = A\cos 3x + B\sin 3x + \dfrac{2}{3}x\sin 3x + \dfrac{1}{4}\sin x$