SHAPE: y "+ a_{1}y '+ a_{0}y = 0 (a_{0}, a_{1} constants)

Ex: y =

So y '=

and y "=

Substituting in the given equation:

or

() = 0

0 for every x, so we should have = 0, which is an equation of the second degree in the variable , call **characteristic equation. **

The solution of the linear differential equation will depend on the roots 1 and 2.

- 1, 2 real and distinct numbers C1 and C2 are particular EDO solutions and the general solution is y = C1 + C2
- 1 = 2 = (real and equal numbers) EDO's overall solution is y = C1 + C2x
- 1 = a + bi, 2 = a - bi (conjugated complexes: a, b real) the general solution is y = C1 + C2

Ex: y "- 2y '- 15y = 0

Characteristic equation: - 2 - 15 = 0 whose roots are: 1 = 5, 2= -3

General solution: y =

Next: Linear Differential Equations of Order N