Final Stretch 2.0 Advanced Mathematics II

Differential Equations

Non-Homogeneous DEs

  • Expect these in Part A (not in Part B as per exam leaks)
  • General form: Ensure equation uses addition ($M , dx + N , dy$); if subtraction appears, expand negative sign
  • Ignore constants ($C_1, C_2$) when testing linear dependence/independence

Case 1: Linearly Dependent

  • Use ratio method: $$\frac{a_1}{a_2} = \frac{b_1}{b_2}$$
  • Substitute $z = a_2 x + b_2 y$ to reduce to separable DE

Case 2: Linearly Independent

  • Solve simultaneous equations for $x$ and $y$ to eliminate constants
  • Then solve as a Homogeneous DE

Bernoulli DEs

  • Coefficient of $\frac{dy}{dx}$ must be 1 before starting
  • Procedure:
    1. Divide by $y^n$
    2. Substitute $v = y^{1-n}$
    3. Solve as Linear DE using Integrating Factor (I.F.)
  • Warning: Never simplify terms like $\frac{C}{x^{-2}}$ as a new constant $A$ — $x$ is a variable, not constant

Complex Numbers

De Moivre's Theorem

  • Focus on multiple angles (not proof)
  • Power expansion:
    • "Reverse" expansion: express powers like $\cos^3 x$ in terms of multiple angles
    • Use identities: $$z + \frac{1}{z} = 2\cos x \quad \text{and} \quad z - \frac{1}{z} = 2i\sin x$$
    • Use Pascal's Triangle for coefficients

Trigonometry of Complex Numbers

  • Exponential forms: $$\cos x = \frac{e^{ix} + e^{-ix}}{2}, \quad \sin x = \frac{e^{ix} - e^{-ix}}{2i}$$
  • Tangent shortcut: For $\tan\left(\frac{\pi}{4} - i\right)$, use Compound Angle Formula instead of converting to $e^{ix}$

Locus & Inequalities

  • Dotted lines: $<$ or $>$
  • Solid lines: $\leq$ or $\geq$
  • Distance concept: $$|z - z_1| < |z - z_2|$$ represents region closer to $z_1$ than $z_2$ (perpendicular bisector)