Foundations — Case 1 - two distinct real roots

This page unpacks every symbol the parent note (Case 1) throws at you, starting from things a 12-year-old already knows, and stacking one idea on top of the last. Nothing below is used before it is built.

1. A function $y$ and the variable $x$

The picture: a smooth line above a horizontal axis. Drag $x$; the dot rides up and down the curve.

Why the topic needs it: the whole chapter is about finding which curve $y(x)$ obeys a rule. Everything else is machinery for pinning down that one curve.

2. The derivative $y^{\prime }$ — how steep the curve is

Look at the amber line touching the curve in the figure: it is the tangent, and $y^{\prime }$ is exactly its steepness. Where the curve is going up, the tangent tilts up; where it flattens at the top, the tangent is horizontal and $y^{\prime }=0$.

Why the topic needs both: the ODE $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$ mixes the height $y$, the slope $y^{\prime }$, and the bend $y^{\prime \prime }$ and demands they cancel. To play, you must know all three.

3. The exponential $e^{rx}$ — the self-copying function

The magic property, which the whole topic leans on: $\frac{d}{dx}{e}^{rx}=r{e}^{rx}.$

The figure shows three exponentials. Notice: $r>0$ (cyan) shoots upward, $r<0$ (amber) decays toward zero, and $r=0$ is the flat line $e^0=1$. The sign of $r$ is the long-term fate of the solution.

4. The letters $a,b,c,r$ — constants vs. unknown

Why the topic needs it: $a$ must be non-zero ($a\ne 0$) or there is no $y^{\prime \prime }$ term and it stops being a second-order equation. The reader must keep straight which symbols are knobs set by the problem and which one we are hunting.

5. The characteristic equation $a{r}^2+br+c=0$

Symbol-by-symbol: $r^2$ came from $y^{\prime \prime }$, $r$ from $y^{\prime }$, the constant term from $y$. (Deep detail in Characteristic equation.)

6. The quadratic formula and the discriminant $b^2-4ac$

The parabola $a{r}^2+br+c$ in the figure shows all three: it crosses the axis twice ($\Delta >0$, our case), touches it once ($\Delta =0$), or misses it entirely ($\Delta <0$). The crossing points are the roots.

Why the topic needs it: "Case 1" is defined by $\Delta >0$. Without this quantity you cannot tell which case you are in.

7. Linear combination and the constants $C_1,C_2$

Why the topic needs it: the general solution must carry $C_1,C_2$; dropping them keeps only one curve out of an infinite family (a classic mistake flagged in the parent note).

8. Linear independence and the Wronskian $W$

Because $r_1\ne r_2$ in Case 1, the factor $(r_2-r_1)\ne 0$ and the exponential is never zero, so $W\ne 0$ — the two exponentials are independent. (Full story: Wronskian and linear independence.)

Why the topic needs it: it is the proof that $C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$ really captures all solutions, not just some.

Prerequisite map

Equipment checklist

Test yourself — cover the right side.