Solve equations with real, repeated, or complex roots. Apply forcing terms and initial conditions accurately. See plots, tables, and exports for stronger problem insight.
The solver handles linear second order equations with constant coefficients and common forcing types.
| Case | Equation | Initial conditions | Root type | One solved form |
|---|---|---|---|---|
| Exponential resonance | y'' - 3y' + 2y = e^x | y(0)=1, y'(0)=0 | Distinct real roots | y(x) = (1 - x)e^x |
| Repeated root | y'' - 4y' + 4y = 0 | y(0)=2, y'(0)=1 | Repeated root | y(x) = (2 - 3x)e^(2x) |
| Complex roots | y'' + 4y = 0 | y(0)=0, y'(0)=2 | Complex conjugates | y(x) = sin(2x) |
Standard model: a y'' + b y' + c y = f(x)
This page solves linear second order equations with constant coefficients.
Characteristic equation: a r² + b r + c = 0
Its roots determine the complementary solution shape.
If the roots are distinct: yc = C₁e^(r₁x) + C₂e^(r₂x)
If the root repeats: yc = (C₁ + C₂x)e^(rx)
If the roots are complex: yc = e^(αx)[C₁cos(βx) + C₂sin(βx)]
Particular solution method: undetermined coefficients.
The calculator chooses a trial based on the right side. It adds x or x² automatically when resonance occurs.
Initial conditions: the solver substitutes y(x₀) and y'(x₀) to solve for C₁ and C₂.
That produces the specific solution used for evaluation and graphing.
It solves linear second order ordinary differential equations with constant coefficients. The right side can be zero, a constant, an exponential, a sine, a cosine, or a first degree polynomial.
It returns both. The page first shows the complementary and particular pieces, then combines them into a general solution. If you add initial conditions, it also computes a specific solution.
The symbolic result still appears. For the numeric evaluation and graph, the page uses C1 = 0 and C2 = 0 as a neutral fallback so a visible curve can still be plotted.
It detects when the forcing trial duplicates part of the complementary solution. When that happens, the solver multiplies the trial by x or x² before solving for the unknown coefficients.
The discriminant b² − 4ac decides the root type. Positive means distinct real roots, zero means a repeated root, and negative means complex conjugate roots.
Yes. Very large coefficients, wide ranges, or positive exponential growth may create extremely large values. The page marks nonfinite points as overflow rather than forcing a misleading number.
Yes. The exports include the main equation details, root information, solution forms, constants, evaluated values, and a sample output table. They are useful for study notes and worked examples.
No. It is focused on common constant coefficient cases taught in maths courses. It is fast and practical, but it does not cover every possible forcing function or variable coefficient equation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.