Foundations — Case 2 - repeated real root — reduction of order

This page is the ground floor. It assumes you have never seen a derivative, an exponential, or a differential equation. Every symbol in the parent note is built here, one brick at a time.

1. What a function is, and what its graph looks like

Picture it as a curve drawn on a grid. The horizontal axis is the input $x$; the vertical axis is the output $y$. Every point on the curve is one input-output pair.

Figure s01 — a function as a curve. Each vertical dashed line pins an input $x$ to its single output $y$.

Why the topic needs it: the whole chapter is about finding an unknown curve $y(x)$. Everything else — slopes, characteristic equations — is machinery for pinning down which curve it is.

2. The derivative $y^{\prime }$ — slope, defined by a limit

We must make "a small step" exact. Take a step of size $h$ from $x$ to $x+h$. The line joining the two curve-points $(x,y(x))$ and $(x+h,y(x+h))$ — a secant line — has slope $\frac{y(x+h)-y(x)}{h}(\text{rise over run}).$

Figure s02 — the secant becomes the tangent as $h\to 0$. The blue secants pivot toward the pink tangent; the rise-over-run triangle shows exactly what $y^{\prime }$ measures.

Why this tool and not another? We need a way to talk about how fast a quantity changes — velocity, growth, decay. The derivative is precisely the machine that answers "how steep, right here?". Look at the pink tangent line in Figure s02: its steepness is the value of $y^{\prime }$.

3. The second derivative $y^{\prime \prime }$ — curvature, slope-of-the-slope

Figure s03 — curvature has a sign. The blue "valley" bends up ($y^{\prime \prime }>0$), the pink "hill" bends down ($y^{\prime \prime }<0$), the yellow straight line has $y^{\prime \prime }=0$.

Why the topic needs it: the equation is called second-order exactly because $y^{\prime \prime }$ appears. Physically $y^{\prime \prime }$ is acceleration; a spring, a circuit, a swinging pendulum all obey rules relating position ($y$), velocity ($y^{\prime }$) and acceleration ($y^{\prime \prime }$). That relation is our differential equation.

4. The chain rule — differentiating a function of a function

Why the topic needs it: we constantly differentiate $e^{rx}$, $x{e}^{rx}$, etc., where a rate $r$ sits inside. The chain rule is the tool that pulls that inner rate $r$ out front — used immediately in §5.

5. Factorials and the exponential $e^{rx}$ — the function that is its own slope

Now we build $e^x$ from nothing and derive its slope rule with the §2 power rule and §4 chain rule, because the whole topic hangs on it. First, one piece of notation.

Figure s04 — the exponential for three signs of $r$. Pink grows ($r>0$), blue decays ($r<0$), yellow is the flat line $e^0=1$.

• If $r>0$: $e^{rx}$ grows (explodes rightward).
• If $r<0$: $e^{rx}$ decays toward $0$ (settles down) — the pale-blue curve.
• If $r=0$: $e^0=1$, a flat constant line.
• Note $e^{rx}>0$ always — it never hits or crosses zero. This is why we can safely "divide by $e^{rx}$" later.

6. The characteristic equation $a{r}^2+br+c=0$ and its discriminant

When the discriminant is $0$, the $\pm \sqrt{}$ vanishes and both roots merge into the single value $r=-\frac{b}{2a}.$ That collapse is the entire drama of Case 2: two roots became one, so we're one solution short. See Characteristic equation of linear ODEs for the full derivation of this transfer from ODE to quadratic.

7. Dimension, linear independence, and the Wronskian

8. Equipment checklist

Test yourself — you're ready for the parent topic when you can answer every one:

Prerequisite map