Second-Order ODE Solver
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.
Use the Second-Order ODE Solver
ODE Coefficients
Time range [0, Tmax] for the plot.
Quick Examples
No solution yet
Enter coefficients and initial conditions, then click Solve.
Damping Classification
Discriminant D = b² − 4ac
Characteristic Equation
Roots
General Solution Form
Particular Solution (with your initial conditions)
Solution Curve — y(t)
Summary
This tool analytically solves homogeneous second-order linear ODEs of the form ay''+by'+cy=0 with constant coefficients. It computes the characteristic equation ar²+br+c=0, finds its roots, classifies the system as overdamped, critically damped, or underdamped, and constructs the general solution applied to your initial conditions y(0) and y'(0). The solution curve is plotted using Chart.js.
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.