Foundations — Case 3 - complex conjugate roots — Euler's formula connection

This page assumes nothing. Every squiggle the parent note uses is torn open and rebuilt below, in an order where each piece stands on the one before it.

0. What a derivative and a "prime" mean

Before anything, the symbol $y^{\prime }$.

See the prerequisite Characteristic Equation of Linear ODEs for how these primes become the powers of $m$.

1. The equation itself: $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$

The letters:

• $a,b,c\in \mathbb{R}$ means ==$a,b,c$ are real numbers== (ordinary numbers on the number line, symbol $\mathbb{R}$).
• $a\ne 0$ so that a $y^{\prime \prime }$ term genuinely exists (otherwise it's first-order).

2. Turning the equation into a polynomial: $m$

$\pm$ (read "plus or minus") is shorthand for two answers at once — one taking $+$, the other $-$. It appears because a quadratic has two roots.

3. The discriminant $\Delta$ — the fork in the road

4. The imaginary unit $i$ and complex numbers

When $\Delta <0$ we must take the square root of a negative number. No real number squares to a negative, so we invent one.

5. The letters $\alpha$ and $\beta$

6. The exponential $e^x$ and $e^{\alpha x}$

The number $e$ raised to an imaginary power ($e^{i\theta }$) is the magic bridge — built from the Maclaurin Series of exp, sin, cos and unpacked in Euler's Formula and the Unit Circle.

7. Sine, cosine, and the angle $\theta$

8. Superposition, Wronskian, and $C_1,C_2$

Prerequisite map

Equipment checklist

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