Build accurate trials for common nonhomogeneous equations. Check roots, resonance, and forcing patterns with confidence. Export results, compare examples, and follow practical solving steps.
Choose a supported forcing model, enter the differential equation coefficients, and estimate a particular solution using the undetermined coefficients method.
| Example | Equation | Forcing model | Typical trial form |
|---|---|---|---|
| 1 | y″ − 3y′ + 2y = 4e^x | Exponential | xAe^x because e^x resonates |
| 2 | y″ + y = 5x − 2 | Polynomial | Ax + B |
| 3 | y″ + 4y = 3sin(2x) | Trigonometric | x[A sin(2x) + B cos(2x)] |
| 4 | y″ − 2y′ + y = e^x(3 + 2x) | Exponential-polynomial | x^2e^x(A + Bx) |
| 5 | y″ + 2y′ + 5y = e^(−x)[2sin(2x)+5cos(2x)] | Exponential-trigonometric | e^(−x)[A sin(2x) + B cos(2x)] |
The method solves linear equations of the form ay″ + by′ + cy = f(x), where f(x) belongs to a supported forcing family.
First solve the characteristic equation ar² + br + c = 0 to get the complementary solution yc.
Next choose a trial particular form yp that matches the forcing model: polynomial, exponential, trigonometric, exponential-polynomial, or exponential-trigonometric.
If any trial term duplicates a complementary term, multiply the entire trial function by x or x² until linear independence is restored.
This calculator then estimates unknown coefficients by applying L[y] = ay″ + by′ + cy to each trial basis term and solving the resulting linear system.
It handles second-order linear differential equations with constant coefficients and supported forcing terms suited to the undetermined coefficients method.
It supports polynomial, exponential, sine-cosine, exponential-polynomial, and exponential-trigonometric inputs. These cover many textbook nonhomogeneous examples.
Resonance means a trial term already appears in the complementary solution. The calculator multiplies the trial by x or x² to restore independence.
The tool builds and solves a linear system from the operator applied to each trial basis term. That gives reliable coefficients for supported models.
No. The method shown here is designed for constant-coefficient equations. Variable-coefficient problems usually need different techniques.
A singular system means the chosen data produced dependent trial equations or unstable matching points. Adjusting coefficients or forcing type usually resolves it.
It provides a practical check at selected x values. It is a verification aid, not a formal symbolic proof.
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.