Further Maths Modelling with differential equations Practice Questions

Compare the model with the SHM equation

Reading off $\omega^{2}$ is the only information the period needs.

Find the angular frequency

Take the positive square root; $\omega$ is a rate, so it is positive.

State the period

This is the exact time for one complete oscillation, in seconds.

Find the natural frequency of the undamped model

Setting the forcing term to zero leaves the free oscillation of the system.

Read off the natural angular frequency

This is the frequency at which the system oscillates when left alone.

State the condition for resonance

There is no damping, so a forcing term at the natural frequency is already a solution of the homogeneous equation.

State the resonant frequency

At this forcing frequency the amplitude of the response grows without bound.

Form the auxiliary equation

Substituting $x=e^{mt}$ into the homogeneous equation turns it into a quadratic in $m$.

Evaluate the discriminant

Its sign is what decides the nature of the damping.

Solve the auxiliary equation

The discriminant is positive, so there are two distinct real roots.

Write down the complementary function

This is the standard form for two distinct real roots.

Note that the complementary function is a transient

The steady state is therefore the particular integral on its own.

Substitute the trial function and compare coefficients

Matching the coefficients of $\cos\left(t\right)$ and $\sin\left(t\right)$ gives two equations.

Select the steady-state solution

This is what the model settles down to once the transient has decayed.

Try a solution of the form $h=e^{mt}$

An exponential reproduces itself under differentiation, which is what makes this work.

Substitute into the model

Every term carries the same factor $e^{mt}$.

Divide by the non-zero factor $e^{mt}$

$e^{mt}$ is never zero, so it can be cancelled.

Note that the coefficients are copied straight across

The order of each derivative becomes the power of $m$.

Read off the three coefficients

They must be taken with their signs exactly as they appear in the model.

Solve the auxiliary equation as a check

These roots are the exponents that appear in the complementary function.

Check the discriminant is consistent with the model

Its sign classifies the damping, so it is a useful check on the coefficients.

Reject any option with the wrong signs

Changing a sign changes the roots and therefore the whole behaviour.

Reject any option with the coefficients swapped

The leading coefficient comes from $\ddot{h}$ and the constant from $h$; they cannot be interchanged.

Reject any option that drops a term

Losing the constant term would turn the quadratic into a different equation entirely.

Select the auxiliary equation

This is the quadratic whose roots give the exponents in the solution.

Form the auxiliary equation

Trying $x=e^{mt}$ turns the model into a quadratic in $m$ whose coefficients involve $k$.

State the condition for critical damping

Critical damping is exactly the boundary case of a repeated root.

Write down the discriminant

Read $a$, $b$ and $c$ straight off the auxiliary equation.

Set the discriminant to zero and solve

The damping constant is positive, so the negative root is rejected.

Substitute the critical value back into the auxiliary equation

With this value of $k$ the quadratic should collapse to a perfect square.

Factorise the auxiliary equation

A perfect square is exactly what a repeated root means.

Check the repeated root

Both roots coincide, which is the definition of critical damping.

Write down the resulting motion

The extra factor of $t$ is what a repeated root contributes.

Confirm the motion has no oscillation

A linear factor has only one root, so the model crosses equilibrium at most once.

Test a smaller damping constant

Halving $k$ makes the discriminant negative, so the model would oscillate.

Test a larger damping constant

Doubling $k$ makes the discriminant positive, so the model would be heavily damped.

Note what happens on either side of the critical value

Below the critical value the model oscillates; above it, it creeps back.

Interpret the answer physically

Critical damping is what a designer wants when overshoot must be avoided.

Recall the auxiliary equation

Trying $x=e^{mt}$ turns the differential equation into a quadratic in $m$.

Recall how the discriminant classifies the damping

The nature of the roots decides whether the model oscillates on its way to rest.

State the critical damping constant

This is the unique positive value of $k$ that gives critical damping.