Interactive Differential Equations Explorer

First-Order Linear Differential Equation

Thermal, RC Circuit, or Fluid Storage Dynamics

Governed by first-order continuous energy accumulation. The response $y(t)$ smoothly approaches forcing input $u(t)$ with continuous time constant $\tau$.

τ y'(t) + y(t) = u(t)

Parameters & Input

Time Constant (τ): 1.00 s

Controls system inertia and response speed

Initial Condition (y₀): 0.00

Input Forcing Function u(t):

Input Amplitude (A): 1.00

Frequency (ω rad/s): 1.00

Simulation Time (T_max): 10.0 s

Time Constant (τ) 1.00 s

63.2% Response Time

Settling Time (~4τ) 4.00 s

Reaches 98% Steady-State

Current Value y(t) 0.00

At synchronized cursor

Time Domain Response

Input u(t) Output y(t)

t = 0.00s

Second-Order System Dynamics

Mass-Spring-Damper, RLC Circuit, & Mechanical Oscillators

Exhibits inertia, dissipation, and elastic store. Exhibits complex transient regimes ranging from harmonic oscillation to heavy overdamping.

m y''(t) + c y'(t) + k y = F(t)

System Parameters

Mass (m): 1.00 kg

Damping Coefficient (c): 0.50 Ns/m

Spring Constant (k): 4.00 N/m

Initial State Variables

y₀ (Position):

v₀ (Velocity):

External Forcing Function F(t)

Force Amplitude (F₀): 2.0

Forcing Frequency (ω): 2.0 rad/s

Natural Freq (ωₓ) 2.00 rad/s

Damping Ratio (ζ) 0.125

Damped Freq (ωₑ) 1.98 rad/s

Damping Regime Underdamped

Physical System Animation

Mass-Spring-Damper Mechanism

Response y(t) & v(t)

y(t) v(t)

2D Phase Portrait (v vs y)

State Trajectory

t = 0.00s