Second-Order Linear DE (Constant Coeff.)

a y^{''} + b y^{'} + cy = 0, r = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Characteristic roots classify overdamped (two real), critically damped (repeated), and underdamped (complex) free response — the same pattern as a 1-DOF oscillator in dynamics.

$a, b, c$: Constant coefficients
$r$: Characteristic root

Step-by-step solved example

y''+5y'+6y=0. Find the general solution.

Free oscillator: the same time axis as kinematics, now as y(t).

1. Roots
r²+5r+6=0 → (r+2)(r+3)=0, r=−2,−3.
$r = - 2, - 3$
2. Solution
y=C1 e^{−2t}+C2 e^{−3t}.
$y = C_{1} e^{- 2 t} + C_{2} e^{- 3 t}$

Answer: y=C1 e^{−2t}+C2 e^{−3t}

10 practice questions

0/10 correct

1.Repeated root r=−4 gives y =

2.y''+ω²y=0 solutions are

3.Discriminant b²−4ac<0 means

4.Particular solution of y''+y=2 is

5.y''−y=0 roots are

6.Initial y(0), y'(0) fix

7.A first-order system equivalent uses

8.Overdamped free motion

9.y''+2y'+y=0 is

10.Forcing sinΩt on an undamped oscillator at Ω=ω causes