Six original Cambridge-style questions on complementary functions: the three cases of the auxiliary equation, finding the arbitrary constants from initial conditions, and describing the motion that a solution represents.
Find the general solution of \(\dfrac{d^{2}y}{dx^{2}} - 7\dfrac{dy}{dx} + 10y = 0\).
Find the general solution of \(9\dfrac{d^{2}y}{dx^{2}} - 12\dfrac{dy}{dx} + 4y = 0\).
Find the general solution of \(\dfrac{d^{2}y}{dx^{2}} + 6\dfrac{dy}{dx} + 34y = 0\).
Find the solution of \(\dfrac{d^{2}y}{dx^{2}} + 2\dfrac{dy}{dx} - 3y = 0\) for which \(y = 4\) and \(\dfrac{dy}{dx} = 0\) when \(x = 0\). Describe the behaviour of \(y\) as \(x \to \infty\).
The displacement \(x\) cm of a particle at time \(t\) s satisfies \(\dfrac{d^{2}x}{dt^{2}} + 4\dfrac{dx}{dt} + 20x = 0\). When \(t = 0\), \(x = 0\) and \(\dfrac{dx}{dt} = 8\).
(a) Find \(x\) in terms of \(t\).
(b) Describe the motion and find the time at which the particle first returns to \(x = 0\).
The constant \(k\) satisfies \(k \ge 0\) and \(\dfrac{d^{2}y}{dx^{2}} + k\dfrac{dy}{dx} + 9y = 0\).
(a) Find the value of \(k\) for which the auxiliary equation has a repeated root, and write down the general solution in that case.
(b) Write down the general solution when \(k = 0\) and when \(k = 10\).
(c) Show that for \(0 < k < 6\) every non-zero solution is a damped oscillation.
Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.