Second Order ODE Calculator
Work out second order ode instantly with clear inputs, formula shown and shareable results.
About Second Order ODE Calculator
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.
Formulas
Auxiliary equation
ar² + br + c = 0
Complex case
roots α ± βi give e^(αx)(C₁cos βx + C₂sin βx)
Worked example
With the default values below, Second Order ODE Calculator returns Solution type of Overdamped — two distinct real roots, solution C₁e^(r₁x) + C₂e^(r₂x). Change any input above and every figure here is recalculated in your browser.
Inputs used
- Coefficient a in a·y'' + b·y' + c·y = 0
- 1
- Coefficient b
- 5
- Coefficient c
- 6
Results
- Solution type
- Overdamped — two distinct real roots, solution C₁e^(r₁x) + C₂e^(r₂x)
- Discriminant b² − 4ac
- 1.00000000
- First root (or real part)
- -2.00000000
- Second root (or imaginary part)
- -3.00000000
Step by step
- 1Values used
- 2Auxiliary equation
- 3Complex case
- 4Solution type= Overdamped — two distinct real roots, solution C₁e^(r₁x) + C₂e^(r₂x)
- 5Discriminant b² − 4ac= 1.00000000
- 6First root (or real part)= -2.00000000
- 7Second root (or imaginary part)= -3.00000000
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
For y'' + 5y' + 6y = 0?
The discriminant is 1, so the roots are −2 and −3 and the solution is C₁e⁻²ˣ + C₂e⁻³ˣ.
Where does this arise physically?
In mass-spring-damper systems and RLC circuits, where the three cases are exactly over, critical and under damping.