Calculator Form
Example Data Table
| Example | Equation | Initial Data | Impulse | Use |
|---|---|---|---|---|
| Spring impulse | y'' + 0.8y' + 4y = 5δ(t - 2) | y(0)=0, y'(0)=0 | A=5, τ=2 | Sudden force response |
| Decay impulse | x' + 1.5x = 3δ(t - 1) | x(0)=0.5 | A=3, τ=1 | Signal shock model |
| Weak damping | 2y'' + 0.2y' + 5y = 4δ(t - 3) | y(0)=1, y'(0)=0 | A=4, τ=3 | Oscillation study |
Formula Used
x'(t) + λx(t) = Aδ(t - τ), x(0)=x₀
x(t)=x₀e^(-λt)+AH(t-τ)e^(-λ(t-τ))
my'' + cy' + ky = Aδ(t - τ), y(0)=y₀, y'(0)=v₀
y(t)=yₕ(t)+A h(t-τ)H(t-τ)
For the second order case, displacement remains continuous. Velocity jumps by A/m.
The calculator uses the characteristic roots of mr² + cr + k = 0.
How to Use This Calculator
- Select first order or second order impulse IVP.
- Enter the initial values and equation coefficients.
- Enter impulse strength A and impulse time τ.
- Set graph range, sample points, and precision.
- Press Calculate to view jumps, roots, values, and graph.
- Use CSV or PDF buttons to save the result.
Article: Dirac Delta Initial Value Problems
What the Delta Term Means
A Dirac delta term models a sudden input. It is not an ordinary function. It acts like an ideal impulse at one exact time. In an initial value problem, this impulse changes the solution instantly. The change depends on the equation order and the impulse strength.
Why IVP Calculations Need Care
Normal forcing terms act over an interval. A delta term acts at a point. This creates a jump condition. For a first order equation, the state usually jumps. For a second order mechanical equation, displacement stays continuous, but velocity jumps. This calculator applies those rules directly.
Laplace Transform Idea
The Laplace transform is a common method for these problems. It converts derivatives into algebraic terms. It also changes δ(t - τ) into an exponential factor. That factor delays the impulse response. The final answer is a sum of the free response and the shifted impulse response.
Second Order Motion
The second order model can describe springs, dampers, beams, and shock systems. The roots of the characteristic equation show the response type. Real roots give non-oscillating motion. A repeated root gives critical damping. Complex roots give oscillation with decay or growth.
Graph and Export Benefits
The graph helps show the effect of the impulse time. Before the impulse, the curve follows only the natural response. After the impulse, a shifted response is added. The table gives sampled values. CSV export helps with spreadsheets. PDF export helps with reports, assignments, and records.
FAQs
1. What is a Dirac delta input?
It is an ideal impulse acting at one instant. It has zero width and finite total strength.
2. What does impulse strength mean?
Impulse strength is the total area of the delta input. It controls the size of the jump.
3. Why does velocity jump in second order problems?
The delta force changes momentum instantly. Displacement stays continuous, but velocity changes by A divided by m.
4. Does the calculator support first order equations?
Yes. Select the first order option and enter λ, x(0), impulse strength, and impulse time.
5. What is τ in the calculator?
τ is the impulse time. The delta input acts at t = τ and changes the solution there.
6. What does H(t-τ) mean?
It is the unit step factor. It keeps the impulse response inactive before the impulse time.
7. Can this solve damped oscillator problems?
Yes. Use the second order form with mass, damping, stiffness, initial displacement, and initial velocity.
8. Why use more graph points?
More points give a smoother curve and better peak estimates. They also create a larger export table.