Particular Solution Calculator
For 2nd Order Linear Non-Homogeneous Differential Equations
This calculator finds the particular solution yp(x) for an equation of the form: ay” + by’ + cy = Dx² + Ex + F.
Value is unitless.
Value is unitless.
Value must be non-zero for this method.
The non-homogeneous part g(x).
The non-homogeneous part g(x).
The non-homogeneous part g(x).
Dynamic Chart of the Solution
What is a Particular Solution Calculator?
A particular solution calculator is a tool designed to find a specific solution to a differential equation that satisfies the equation without any arbitrary constants. This contrasts with the “general solution,” which includes arbitrary constants (like C1, C2) and represents a family of all possible solutions. This calculator focuses on a common type of problem: finding the particular solution for a second-order linear non-homogeneous differential equation with constant coefficients.
Specifically, it solves equations in the form ay'' + by' + cy = g(x), where g(x) is a polynomial. Engineers, physicists, mathematicians, and students use this to model system responses to external forces or inputs, where g(x) represents that external input.
Particular Solution Formula and Explanation
To find the particular solution for the equation ay'' + by' + cy = Dx² + Ex + F, we use the Method of Undetermined Coefficients. The core idea is to guess that the particular solution, yp(x), will have a similar form to the function g(x) on the right-hand side.
Since our g(x) is a quadratic polynomial, we assume the particular solution is also a quadratic polynomial:
yp(x) = Ax² + Bx + C
We then find the first and second derivatives of our guess:
y'p(x) = 2Ax + By''p(x) = 2A
Next, we substitute these back into the original differential equation and group terms by powers of x. By equating the coefficients of each power of x on both sides, we get a system of linear equations to solve for A, B, and C. You can find more details at a resource like the Linear Algebra Solver.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a, b, c | Coefficients of the homogeneous part of the equation | Unitless | Any real number |
| D, E, F | Coefficients of the polynomial non-homogeneous term g(x) | Unitless | Any real number |
| A, B, C | The undetermined coefficients we solve for in yp(x) | Unitless | Calculated based on a, b, c, D, E, F |
Practical Examples
Example 1: Standard Case
Let’s solve the equation: y'' + 3y' + 2y = 4x²
- Inputs: a=1, b=3, c=2, D=4, E=0, F=0
- Process: We assume yp = Ax² + Bx + C. After substituting and solving, we find:
A = 4 / 2 = 2B = (0 - 3 * 2 * 2) / 2 = -6C = (0 - 1 * 2 * 2 - 3 * -6) / 2 = 7
- Result: The particular solution is
yp(x) = 2x² - 6x + 7.
Example 2: No y’ or y terms
Let’s solve the equation: 2y'' = 6x² + 4x
- Inputs: a=2, b=0, c=0. Note: This calculator requires c ≠ 0. This case would require a different assumed form (
Ax⁴ + Bx³ + Cx²) which you can learn about in our guide to the Method of Undetermined Coefficients.
How to Use This Particular Solution Calculator
- Identify Coefficients: Look at your differential equation
ay'' + by' + cy = Dx² + Ex + Fand identify the values for a, b, c, D, E, and F. - Enter Values: Input these six coefficients into the corresponding fields. The inputs are all unitless numbers.
- Handle Special Cases: This calculator requires the coefficient ‘c’ to be non-zero. If c=0, the method changes, and you’ll need to consult a textbook or a more advanced Differential Equation Solver.
- Calculate: Click the “Calculate Particular Solution” button.
- Interpret Results: The calculator will display the determined coefficients A, B, and C, and the final particular solution
yp(x). The chart will also update to show a graph of this solution.
Key Factors That Affect the Particular Solution
- The form of g(x): The structure of the function on the right side dictates the entire method. If g(x) were an exponential or sine function, our initial guess for yp would be completely different.
- The value of ‘c’: As seen in this calculator, if the coefficient ‘c’ is zero, the standard guess of Ax² + Bx + C is insufficient, and the polynomial degree of the guess must be increased.
- The value of ‘b’: If both ‘c’ and ‘b’ are zero, the guess must be modified again.
- Relationship to the Homogeneous Solution: A major complication arises if the assumed form of yp is also a solution to the homogeneous equation (
ay'' + by' + cy = 0). In that case, the guess must be multiplied by x (or x²) to find a valid solution. This is a crucial concept explored in our Homogeneous Equations guide. - Initial Conditions: While this calculator finds the particular solution form, initial conditions (like y(0)=1, y'(0)=0) are needed to find the constants in the *general* solution, which is y(x) = yc(x) + yp(x).
- Coefficient Magnitudes: The relative sizes of a, b, and c determine the damping and natural frequency of the system, which influences how the particular solution behaves in the context of the overall response.
Frequently Asked Questions (FAQ)
What’s the difference between a general and particular solution?
A general solution includes arbitrary constants and represents all possible solutions. A particular solution is a single, specific solution with no arbitrary constants, obtained by either applying initial conditions or, as in this calculator, by finding a function that satisfies the non-homogeneous part.
What units do the coefficients have?
In this abstract mathematical context, all coefficients (a, b, c, D, E, F) and the variable x are considered unitless. If the differential equation were modeling a physical system, they would have units (e.g., mass, damping coefficient, spring constant).
What if the function on the right side, g(x), isn’t a polynomial?
The Method of Undetermined Coefficients also works if g(x) is an exponential function (e.g., kerx), a sine or cosine function (e.g., k*sin(mx)), or a combination of these. However, the assumed form of yp changes accordingly. You can’t use this specific calculator for those forms.
Why does the calculator require c ≠ 0?
If c = 0, the term c * (Ax² + Bx + C) becomes zero. This means the coefficient of x², which is c*A, is zero, making it impossible to solve c*A = D if D is not zero. The assumed form for yp must be modified (e.g., to x(Ax² + Bx + C)).
Is the particular solution unique?
No, there are infinitely many particular solutions. However, the Method of Undetermined Coefficients provides the simplest polynomial form. Any other particular solution can be found by adding a term from the homogeneous solution to the one we found. For a deeper dive, see our article on Second Order ODEs.
Does this calculator give the complete answer to a differential equation?
No. It provides only the particular solution, yp. The complete general solution is y(x) = yc(x) + yp(x), where yc(x) is the solution to the corresponding homogeneous equation.
How are the intermediate coefficients A, B, and C calculated?
They are found by solving a system of linear equations derived from substituting yp into the ODE: 1) cA = D, 2) 2bA + cB = E, 3) 2aA + bB + cC = F.
What does the graph represent?
The graph shows a plot of the function yp(x) that you calculated. It’s a visual representation of how the particular solution behaves over a range of x values, giving you insight into its shape and trajectory.