Find Particular Solution To Differential Equation Calculator

Easily find the particular solution (y_p) for second-order linear non-homogeneous differential equations of the form ay" + by' + cy = Ae^rx using the method of undetermined coefficients with this calculator.

Calculator: ay" + by' + cy = Ae^rx

What is a Particular Solution to a Differential Equation Calculator?

A particular solution to differential equation calculator is a tool designed to find a specific solution, y_p(x), to a non-homogeneous linear differential equation, typically of the form ay" + by' + cy = f(x). Unlike the complementary solution (y_c(x)), which solves the homogeneous part (ay" + by' + cy = 0), the particular solution directly addresses the forcing function f(x) on the right-hand side.

This calculator specifically uses the method of undetermined coefficients for cases where f(x) is of the form Ae^rx. It helps you determine the form of y_p(x) and calculate the unknown coefficients based on the values of a, b, c, A, and r.

This calculator is useful for students learning differential equations, engineers, physicists, and anyone working with models described by linear differential equations with constant coefficients and a specific type of forcing function. A common misconception is that the particular solution is the complete solution; however, the general solution is y(x) = y_c(x) + y_p(x).

Particular Solution Formula and Mathematical Explanation (Method of Undetermined Coefficients for f(x)=Ae^rx)

We are looking for a particular solution y_p(x) to the equation:

ay" + by' + cy = Ae^rx

The method of undetermined coefficients involves guessing a form for y_p(x) based on f(x), substituting it into the differential equation, and solving for the unknown coefficients.

Step 1: Consider the Characteristic Equation

The characteristic equation of the homogeneous part is am² + bm + c = 0. Its roots determine the form of the complementary solution and influence the form of the particular solution.

Step 2: Guess the Form of y_p(x)

Based on f(x) = Ae^rx, our initial guess for y_p(x) is Ce^rx. However, we need to compare 'r' with the roots of the characteristic equation:

• Case 1: 'r' is NOT a root of am² + bm + c = 0. The form of y_p(x) is Ce^rx. Substituting into the DE gives C(ar² + br + c)e^rx = Ae^rx, so C = A / (ar² + br + c).
• Case 2: 'r' IS a single root of am² + bm + c = 0. This means ar² + br + c = 0, but 2ar + b ≠ 0. The form of y_p(x) is Cxe^rx. Substituting gives C(2ar + b)e^rx = Ae^rx, so C = A / (2ar + b).
• Case 3: 'r' IS a double root of am² + bm + c = 0. This means ar² + br + c = 0 and 2ar + b = 0. The form of y_p(x) is Cx²e^rx. Substituting gives 2aCe^rx = Ae^rx, so C = A / (2a).

Variables Table:

Variable	Meaning	Unit	Typical Range
a, b, c	Coefficients of y", y', and y respectively	Varies	Real numbers, a ≠ 0
A	Coefficient of e^rx in f(x)	Varies	Real numbers
r	Exponent in e^rx	Varies	Real numbers
m	Roots of the characteristic equation	Varies	Real or Complex numbers
C	Undetermined coefficient in yp(x)	Varies	Real number
yp(x)	Particular solution	Varies	Function of x

The particular solution to differential equation calculator automates these checks and calculations.

Practical Examples

Example 1: 'r' is not a root

Consider y" + 3y' + 2y = 6e^x. Here a=1, b=3, c=2, A=6, r=1.

Characteristic equation: m² + 3m + 2 = 0 => (m+1)(m+2)=0 => m=-1, m=-2.

'r'=1 is not a root. So, y_p = Ce^x. ar²+br+c = 1(1)² + 3(1) + 2 = 1 + 3 + 2 = 6. C = A/6 = 6/6 = 1. So, y_p(x) = e^x. The calculator would find this.

Example 2: 'r' is a single root

Consider y" – y = 2e^x. Here a=1, b=0, c=-1, A=2, r=1.

Characteristic equation: m² – 1 = 0 => m=1, m=-1.

'r'=1 is a single root. So, y_p = Cxe^x. 2ar+b = 2(1)(1) + 0 = 2. C = A/2 = 2/2 = 1. So, y_p(x) = xe^x. The particular solution to differential equation calculator handles this case.

How to Use This Particular Solution to Differential Equation Calculator

1. Enter Coefficients a, b, c: Input the coefficients of y", y', and y from your differential equation. Ensure 'a' is not zero.
2. Enter Forcing Function Parameters A, r: Input the values of 'A' and 'r' from your forcing function f(x) = Ae^rx.
3. Calculate: Click the "Calculate" button or just change input values. The calculator automatically updates.
4. Review Results:
   • Primary Result: Shows the calculated particular solution y_p(x).
   • Intermediate Results: Displays the roots of the characteristic equation, the values of ar²+br+c and 2ar+b to determine the case, the form of y_p, and the value of the coefficient C.
5. Reset or Copy: Use "Reset" to go back to default values or "Copy Results" to copy the solution details.

The output helps you understand how the form of y_p was chosen and how C was calculated.

Key Factors That Affect Particular Solution Results

• Coefficients a, b, c: These determine the roots of the characteristic equation, which crucially affects the form of y_p if 'r' matches one of the roots.
• Value of 'r': Whether 'r' is a root of am²+bm+c=0 (and its multiplicity) dictates the form of y_p (Ce^rx, Cxe^rx, or Cx²e^rx).
• Value of 'A': This directly scales the undetermined coefficient C.
• Roots of the Characteristic Equation: The nature and values of these roots (real distinct, real repeated, complex) determine the complementary solution and are compared with 'r'.
• Type of Forcing Function f(x): This calculator is for f(x)=Ae^rx. Different forms of f(x) (like sines, cosines, polynomials) require different initial guesses for y_p, as covered by the method of undetermined coefficients.
• Initial Conditions (for general solution): While not used for y_p directly, initial conditions are needed to find the constants in the complementary solution y_c when forming the general solution y = y_c + y_p and solving an initial value problem.

Frequently Asked Questions (FAQ)

What is the difference between a particular solution and a general solution?

A particular solution (y_p) is ANY one function that satisfies the non-homogeneous equation ay"+by'+cy=f(x). The general solution is y = y_c + y_p, where y_c is the complementary solution (solving ay"+by'+cy=0) containing arbitrary constants, and y_p is one particular solution.

What is the complementary solution?

The complementary solution (y_c) is the general solution to the corresponding homogeneous equation ay"+by'+cy=0. It's found using the roots of the characteristic equation.

Why does the form of y_p change if 'r' is a root?

If 'r' is a root, the simple guess Ce^rx would be part of the complementary solution y_c, making it satisfy ay"+by'+cy=0, not Ae^rx. We multiply by x or x² to get a linearly independent function that can solve the non-homogeneous equation.

Can this calculator handle f(x) = Acos(kx) or Asin(kx)?

This specific calculator is set up for f(x) = Ae^rx. The method of undetermined coefficients can handle trigonometric f(x), but the form of y_p would be Ccos(kx) + Dsin(kx) (or multiplied by x if ik are roots), requiring a different calculation for C and D.

What if 'a' is zero?

If 'a' is zero, the equation becomes a first-order linear differential equation (by'+cy=f(x)), not a second-order one. This calculator assumes a=0.

What if the roots are complex?

The roots of am²+bm+c=0 can be complex. The comparison is still made with 'r'. If 'r' is real, it won't be a complex root unless r=0 and the roots are purely imaginary, but the principle is the same.

How do I find the complete general solution?

First, find the complementary solution y_c using the roots from the characteristic equation solver. Then, use this particular solution to differential equation calculator to find y_p. The general solution is y(x) = y_c(x) + y_p(x).

What if f(x) is a sum of terms?

If f(x) = f1(x) + f2(x), you can find a particular solution yp1 for f1(x) and yp2 for f2(x) separately, and then y_p = yp1 + yp2 (by the superposition principle).