further differential equations

Further Differential Equations extend beyond first-order separable equations to include second-order linear differential equations, homogeneous and non-homogeneous forms, and solution methods using auxiliary equations.

Second-Order Linear Differential Equations take the standard form: $$a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)$$

where a, b, c are constants. When f(x) = 0, the equation is homogeneous; otherwise, it's non-homogeneous.

The Auxiliary Equation Method: For homogeneous equations, substitute $y = e^{mx}$ to obtain the auxiliary equation $am^2 + bm + c = 0$. The discriminant $\Delta = b^2 - 4ac$ determines solution types:

• Two distinct real roots ($m_1, m_2$): General solution is $y = Ae^{m_1x} + Be^{m_2x}$ • Repeated root (m): General solution is $y = (A + Bx)e^{mx}$ • Complex roots ($\alpha \pm \beta i$): General solution is $y = e^{\alpha x}(A\cos\beta x + B\sin\beta x)$

Particular Integral (PI): For non-homogeneous equations, the complete solution = Complementary Function (CF) + Particular Integral. The CF solves the homogeneous part; the PI is any solution satisfying the full equation.

Trial functions for PI depend on f(x):

• Polynomial → try polynomial of same degree
• $e^{kx}$ → try $\lambda e^{kx}$
• $\sin kx$ or $\cos kx$ → try $\lambda\cos kx + \mu\sin kx$

Cambridge Note: Always verify your PI doesn't duplicate CF terms; if so, multiply by x.

Boundary/Initial Conditions: Use given values of y and dy/dx at specific points to determine constants A and B in the general solution.

Differential equations model dynamic systems where rates of change interact. Second-order equations naturally arise when acceleration (second derivative) relates to position.

Simple Harmonic Motion (SHM): A mass-spring system follows $\frac{d^2x}{dt^2} = -\omega^2 x$. The auxiliary equation $m^2 + \omega^2 = 0$ gives complex roots $\pm\omega i$, yielding $x = A\cos\omega t + B\sin\omega t$ — oscillatory motion! The mnemonic COSSIN reminds us: Complex roots produce Oscillating Solutions with SINe and cosine.

Damped Oscillations: Adding resistance (like a shock absorber) gives $\frac{d^2x}{dt^2} + 2k\frac{dx}{dt} + \omega^2 x = 0$. The discriminant determines behavior:

• Overdamped ($k^2 > \omega^2$): Two real roots, no oscillation
• Critically damped ($k^2 = \omega^2$): Repeated root, fastest return to equilibrium
• Underdamped ($k^2 < \omega^2$): Complex roots, decaying oscillations

Electrical Circuits (RLC): Current I in a circuit with resistance R, inductance L, capacitance C satisfies $L\frac{d^2I}{dt^2} + R\frac{dI}{dt} + \frac{1}{C}I = V(t)$. This is mathematically identical to the damped oscillator!

Analogy: Think of solving differential equations like reverse engineering a recipe. You're given how ingredients combine (the equation) and must deduce the original mixture (the solution). Boundary conditions tell you specific measurements that confirm your recipe is correct.

Memory Aid RAD: Real roots give Add exponentials; Duplicate roots add x; Complex gives Damped oscillations.