Real, Distinct Roots

The real, distinct roots case is simply what happens when the characteristic equation has two different ordinary numbers as its roots. All we're really saying here is that the answer is two exponential curves added together, and it's the case you'll see for systems that settle (or blow up) without ever swinging back and forth. In $y = c_1e^{r_1t} + c_2e^{r_2t}$, the pieces are $e^{r_1t}$ and $e^{r_2t}$ (two curves that grow or shrink at their own rates, set by the roots) and $c_1$, $c_2$ (how much of each you use). It's the easiest of the three cases: each root gives you one curve, superposition says any mix of them is a solution, and the starting conditions just tell you the mix.

Steps:

1. Solve the characteristic equation.
2. Write the general solution.
3. Use $y(t_0)$ and $y'(t_0)$ to get two equations in $c_1$ and $c_2$, and solve.

Step 3 is just a 2×2 linear system, the same kind you've solved since algebra.

Long-term behavior. Think about what each exponential does as $t$ grows. A negative root decays, a positive root blows up, and the bigger the root, the faster it moves. So the largest root (as long as its coefficient isn't zero) sets the long-term behavior: if all the roots are negative, the solution decays to 0, and if any root is positive, it blows up. With roots of both signs, the solution grows unless the initial conditions happen to cancel the growing part exactly.

A common mistake. It's tempting to read off $y'(0) = c_2$, as if the second constant were the starting slope. It isn't: differentiate first. $y' = r_1c_1e^{r_1t} + r_2c_2e^{r_2t}$, so $y'(0) = r_1c_1 + r_2c_2$, and both constants show up in both equations.

An overdamped door closer has two negative real roots: the door swings shut smoothly without bouncing.