Repeated Roots Differential Equations Calculator

Differential equations with repeated roots occur when the characteristic equation has roots that are not distinct. This calculator helps solve such equations by finding the general solution and particular solution when necessary.

Introduction

Repeated roots in differential equations occur when the characteristic equation has roots with multiplicity greater than one. These cases require special methods to find the general solution, often involving additional terms that account for the repeated nature of the roots.

This calculator solves second-order linear homogeneous differential equations with repeated roots. It provides the general solution and explains the method used to derive it.

Formula

For a second-order linear homogeneous differential equation with constant coefficients:

ay'' + by' + cy = 0

When the characteristic equation has repeated roots r, the general solution is:

y(x) = (C₁ + C₂x)e^rx

Where C₁ and C₂ are arbitrary constants determined by initial conditions.

How to Use the Calculator

Enter the coefficients a, b, and c from your differential equation.
Click "Calculate" to find the general solution.
Review the solution and the explanation of the method used.

Note: This calculator assumes the characteristic equation has repeated roots. If the roots are distinct, use a different calculator.

Worked Example

Consider the differential equation:

y'' - 6y' + 9y = 0

The characteristic equation is:

r² - 6r + 9 = 0

This factors to (r - 3)² = 0, giving a repeated root r = 3.

The general solution is:

y(x) = (C₁ + C₂x)e^3x

This calculator would provide this solution when given the coefficients a=1, b=-6, c=9.

FAQ

What is a repeated root in differential equations?

A repeated root occurs when the characteristic equation has roots with multiplicity greater than one. This requires special methods to find the general solution.

How do I know if my equation has repeated roots?

Solve the characteristic equation and check if any roots have multiplicity greater than one. If the discriminant is zero, you have repeated roots.

What if my equation has complex repeated roots?

Complex repeated roots are handled similarly, with the solution involving sine and cosine functions. This calculator does not handle complex roots.