Solving the Quadratic
For \(ay'' + by' + cy = 0\), the characteristic equation is \(ar^2 + br + c = 0\). If the roots \(r_1, r_2\) are real and distinct, the general solution is:
\[y = C_1 e^{r_1 x} + C_2 e^{r_2 x}\]
Worked Examples
Example 1: Real Roots
Solve \(y'' - 5y' + 6y = 0\).
- Eq: \(r^2 - 5r + 6 = 0 \implies (r-2)(r-3) = 0\).
- Roots: \(r=2, r=3\).
- Result: \(y = C_1 e^{2x} + C_2 e^{3x}\).
The Bridge to Quantum Mechanics
Real roots in the characteristic equation of the Schrödinger Equation correspond to Forbidden Regions. If the potential \(V\) is greater than the energy \(E\), the roots of the DE are real, and the wavefunction must be an exponential: \(\psi = A e^{\kappa x} + B e^{-\kappa x}\). This describes how a particle's probability "dies out" when it tries to enter a wall. This exponential decay is what makes "Classical Physics" look solid—the electrons just can't penetrate very far into the forbidden regions of other atoms.