What form of ODE does this tool solve?

It solves homogeneous linear second-order ODEs with constant coefficients: ay''+by'+cy=0. Nonhomogeneous (forced) equations and variable-coefficient equations are not supported.

What is the characteristic equation?

Substituting y=e^(rt) into the ODE yields the characteristic equation ar²+br+c=0. The roots r₁, r₂ of this quadratic determine the solution structure.

What do overdamped, critically damped, and underdamped mean?

These describe how quickly the system returns to equilibrium. Overdamped (D>0): two real roots, exponential decay without oscillation. Critically damped (D=0): repeated real root, fastest non-oscillatory return. Underdamped (D<0): complex roots, decaying oscillation.

No. If a=0 the equation reduces to a first-order ODE, which is handled by the First-Order ODE Solver tool.

Are initial conditions required?

Yes. The general solution contains two arbitrary constants C₁ and C₂ that are determined only when y(0) and y'(0) are specified.

How is the solution plotted?

The tool evaluates the exact analytical solution at evenly spaced time points over the range [0, T] and plots the result using Chart.js. You can adjust the plot duration using the Tₘₐₓ field.

Second-Order ODE Solver

Name: Second-Order ODE Solver
Availability: InStock
Author: Nham Vu

Enter coefficients a, b, c for ay''+by'+cy=0 with initial conditions to get the exact analytical solution, damping classification, and a solution curve.

ODE Coefficients

a·y″ + b·y′ + c·y = 0

a (y″)

b (y′)

c (y)

y(0)

y′(0)

Plot duration T_max

Time range [0, T_max] for the plot.

Quick Examples

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Enter coefficients and initial conditions, then click Solve.

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Summary

Enter coefficients a, b, c for ay''+by'+cy=0 with initial conditions to get the exact analytical solution, damping classification, and a solution curve.

How it works

Enter the coefficients a, b, and c for the equation ay''+by'+cy=0.
Set the initial conditions: y(0) and y'(0) (the derivative at t=0).
The tool solves the characteristic equation ar²+br+c=0 using the quadratic formula.
The discriminant D=b²-4ac determines the damping case: D>0 overdamped, D=0 critically damped, D<0 underdamped.
Constants C₁ and C₂ are found by applying the initial conditions.
The exact solution y(t) is displayed in symbolic form and plotted over a chosen time range.

Use cases

Solve spring-mass-damper systems in mechanical engineering (my''+cy'+ky=0).
Analyze RLC circuit transient responses described by Lq''+Rq'+q/C=0.
Study damped harmonic oscillators in physics courses.
Verify hand-calculated solutions for differential equations homework.
Explore how changing damping affects oscillation behavior.
Quickly classify systems as overdamped, critically damped, or underdamped.
Visualize transient decay versus oscillatory behavior in control systems.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu