Calculator Inputs
This solver targets first-order linear nonhomogeneous equations with constant coefficient and mixed forcing terms.
Formula Used
The calculator solves the supported first-order linear model below.
y′ + ay = Aeekt + Assin(ωt) + Accos(ωt) + Lt + C0
The integrating factor is μ(t) = eat. Multiplying the equation by μ(t) gives an exact derivative.
d/dt [eaty] = eatQ(t)
The homogeneous solution is yh(t) = Ce-at when a ≠ 0, and yh(t) = C when a = 0.
The particular solution is split into three addable parts:
- Exponential forcing: yp,exp(t) = Aeekt/(a + k), or Aete-at when k = -a.
- Trigonometric forcing: use yp,trig(t) = Msin(ωt) + Ncos(ωt), where M and N come from matching coefficients.
- Linear forcing: use yp,poly(t) = αt + β when a ≠ 0, or 0.5Lt2 + C0t when a = 0.
The complete answer is y(t) = yh(t) + yp(t). The initial condition determines the constant C.
How to Use This Calculator
- Enter the coefficient a from the left side of the equation.
- Fill the forcing amplitudes and rates for exponential, sine, cosine, linear, and constant terms.
- Type the initial point t0 and the initial value y(t0).
- Enter the evaluation point where you want the numeric solution.
- Press Solve Equation to display the result above the form.
- Review the particular parts, constant, full solution, and residual check.
- Use the CSV and PDF buttons to keep a copy of the solution summary.
Example Data Table
| a | Ae | k | As | Ac | ω | L | C0 | t0 | y(t0) | t | y(t) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 6 | 1 | 4 | 1 | 3 | 2 | 5 | 0 | 3 | 1 | 9.286274 |
FAQs
1. What equation type does this calculator solve?
It solves first-order linear nonhomogeneous equations with a constant left-side coefficient and combined exponential, trigonometric, linear, and constant forcing terms.
2. Can it solve higher-order differential equations?
No. This version focuses on first-order linear equations only. Higher-order equations need a different symbolic setup and a different matching strategy.
3. What happens when a equals zero?
The homogeneous part becomes a constant. The forcing terms are then integrated directly, so the calculator switches to the correct simplified formulas automatically.
4. Why is there a residual check?
The residual shows y′ + ay − Q(t) at the evaluation point. A value near zero confirms the displayed solution satisfies the entered equation numerically.
5. What if I do not need one forcing term?
Enter zero for that amplitude. The solver drops that contribution and computes the remaining parts without changing the rest of the workflow.
6. Does the calculator handle resonance?
Yes. When the exponential rate satisfies k = −a, the exponential particular solution changes form to te−at, which the calculator applies automatically.
7. Are the displayed formulas exact?
The underlying computation uses floating-point arithmetic. The displayed expressions are rounded for readability, while the solution logic follows the exact closed-form structure.
8. Why export results to CSV or PDF?
Exports make it easier to store solved cases, compare parameter sets, attach results to coursework, or keep records for technical reviews.