Work out second order ode instantly with clear inputs, formula shown and shareable results.
A homogeneous linear second-order equation reduces to the auxiliary equation ar² + br + c = 0. Its discriminant decides the behaviour: two real roots give exponential decay, a repeated root critical damping, and complex roots oscillation.
Auxiliary equation
ar² + br + c = 0
Complex case
roots α ± βi give e^(αx)(C₁cos βx + C₂sin βx)
The discriminant is 1, so the roots are −2 and −3 and the solution is C₁e⁻²ˣ + C₂e⁻³ˣ.
In mass-spring-damper systems and RLC circuits, where the three cases are exactly over, critical and under damping.