Differential equations

Explanation

• A first-order linear differential equation must first be written as $y'+P(x)y=Q(x)$. Its integrating factor is $I(x)=e^{\int P(x)\,dx}$, which may be quoted.
• Multiplying every term by $I$ makes the left side the product derivative $(Iy)'$, so integration gives $Iy=\int I Q\,dx+C$ and division by $I$ gives the general solution.
• This method is appropriate only when $y$ and $y'$ occur linearly and $P,Q$ depend on the independent variable alone.
• If the coefficient of $y'$ is not $1$, divide through before forming $I$.
• Examiners expect the integrating factor, the product-derivative step and the arbitrary constant to be shown.

Explanation

• A general solution represents a family of solution curves and contains the correct number of independent arbitrary constants: normally one for a first-order equation and two for a second-order equation. A particular solution results when initial or boundary conditions determine all those constants.
• Conditions involving derivatives must be applied to the derivative of the complete general solution.
• The phrase particular integral has a different meaning: it is one trial response to a non-homogeneous forcing term, not a solution selected by initial data.
• The specification also expects sketches of members of a solution family.
• Examiners therefore look for constants in the general solution, correct use of every condition, and curve features such as common asymptotes or the effect of changing the constant.

Explanation

• A differential-equation model begins by defining the independent variable, dependent variable, units and any positive direction. Translate each verbal rate separately before combining them.
• In kinematics, $v=ds/dt$ and $a=dv/dt$; a resistance force acts opposite to the velocity, so its sign follows the chosen direction.
• In mixing, the rate of change is input rate minus output rate, with concentration equal to amount divided by volume.
• After solving, constants, equilibrium values and limiting behaviour must be interpreted in context.
• Examiners also expect modelling assumptions to be recognised, such as constant coefficients, perfect mixing or a resistance law valid over the stated range, and any non-physical prediction to be identified.

Explanation

• For the homogeneous constant-coefficient equation $y''+ay'+by=0$, substitute $y=e^{mx}$ to form the auxiliary equation $m^2+am+b=0$.
• Its two roots determine two independent solution terms.
• Distinct real roots $r,s$ give $Ae^{rx}+Be^{sx}$; a repeated root $r$ gives $(A+Bx)e^{rx}$; conjugate roots $p\pm iq$ give $e^{px}(A\cos qx+B\sin qx)$.
• Initial conditions are imposed only after the complete real general solution and any needed derivatives have been written.
• Examiners expect the auxiliary equation, its roots and the matching form of solution, rather than unsupported answers or unresolved complex exponentials.

Explanation

• For $y''+ay'+by=f(x)$, the general solution is the complementary function plus a particular integral: $y=y_{\mathrm c}+y_{\mathrm p}$. The specification uses forcing terms $ke^{px}$, linear or quadratic polynomials, and $m\cos\omega x+n\sin\omega x$.
• Choose a trial from the same family with enough coefficients to reproduce every term after differentiation.
• If that trial overlaps the complementary function, multiply the whole trial by $x$ until it is independent; a repeated overlap may require $x^2$.
• Substitute to determine the coefficients, combine CF and PI, then apply conditions.
• Examiners expect resonance to be handled explicitly and conditions to be applied to the complete solution.

Explanation

• For the auxiliary equation $m^2+am+b=0$, the discriminant $\Delta=a^2-4b$ determines the form of the differential-equation solution. If $\Delta>0$, two distinct real roots give two exponential terms.
• If $\Delta=0$, the repeated root $r$ gives $(A+Bx)e^{rx}$. If $\Delta<0$, roots $p\pm iq$ give $e^{px}(A\cos qx+B\sin qx)$.
• The imaginary part $q$ produces oscillation; the real part $p$ controls exponential growth or decay.
• Examiners expect parameter ranges to follow from a correctly simplified discriminant and the transition case to use the repeated-root form.
• A negative real root alone does not imply oscillation.

Explanation

• Simple harmonic motion about equilibrium satisfies $x''=-\omega^2x$: acceleration is proportional to displacement and directed towards equilibrium.
• Its general solution is $x=A\cos\omega t+B\sin\omega t$, equivalently $x=R\cos(\omega t+\phi)$.
• The amplitude is $R=\sqrt{A^2+B^2}$, the period is $T=2\pi/\omega$, maximum speed is $\omega R$, and maximum acceleration magnitude is $\omega^2R$.
• At equilibrium the speed magnitude is greatest and acceleration is zero; at an extreme position the speed is zero and acceleration magnitude is greatest.
• Examiners expect initial conditions to determine phase and amplitude and signs to be interpreted using the stated positive direction.

Explanation

• A free damped oscillator can be modelled by $x''+2\lambda x'+\omega_0^2x=0$ with $\lambda>0$, where resistance is proportional to the derivative of displacement. Complex roots give underdamped motion: oscillations within a decaying exponential envelope.
• A repeated negative root gives critical damping, the fastest non-oscillatory return to equilibrium.
• Distinct negative real roots give overdamping and a slower non-oscillatory return.
• A periodic driving force on the right produces forced vibration; the complementary function is the transient response and the particular integral is the persistent steady response.
• Examiners expect classification from the auxiliary roots and an interpretation of decay, oscillation, frequency and long-term motion.

Explanation

• The specified coupled models are linear first-order systems with one independent variable and two dependent variables, of the form $x'=ax+by+f(t)$ and $y'=cx+dy+g(t)$. One method differentiates an equation, eliminates the other dependent variable and solves the resulting second-order equation; the eliminated variable is then recovered from an original equation.
• Initial data may also determine a derivative value through the system.
• Interaction terms must carry consistent signs: a transfer leaving one compartment may enter another.
• Examiners expect both final functions to satisfy both original equations and the model to be interpreted through equilibria, signs and limiting behaviour.
• Predictions such as negative populations mark a limit of the model's validity.