Second Order Differential Equations Repeated Roots Calculator

Second order differential equations with repeated roots occur when the characteristic equation has a double root. This calculator solves such equations and provides the general solution, which includes an additional term involving the natural logarithm function.

Introduction

Second order differential equations with repeated roots are a special case of linear differential equations. When the characteristic equation has a double root, the general solution includes an additional term involving the natural logarithm function. This calculator helps you find the general solution for such equations.

This calculator assumes the differential equation is linear, homogeneous, and has constant coefficients. It does not solve non-linear or non-homogeneous equations.

Formula

For a second order differential equation with repeated roots, the general solution is given by:

y(x) = (C₁ + C₂x)e^rx

Where:

r is the repeated root of the characteristic equation
C₁ and C₂ are arbitrary constants determined by initial conditions

The characteristic equation for the differential equation ay'' + by' + cy = 0 is:

ar² + br + c = 0

When the discriminant (b² - 4ac) is zero, there is a repeated root r = -b/(2a).

Worked Example

Let's solve the differential equation y'' - 4y' + 4y = 0 with initial conditions y(0) = 2 and y'(0) = 5.

Step 1: Find the characteristic equation

The characteristic equation is r² - 4r + 4 = 0.

Step 2: Solve the characteristic equation

The discriminant is 16 - 16 = 0, so there is a repeated root r = 2.

Step 3: Write the general solution

The general solution is y(x) = (C₁ + C₂x)e^2x.

Step 4: Apply initial conditions

Using y(0) = 2: (C₁ + C₂·0)e⁰ = C₁ = 2.

Using y'(0) = 5: The derivative is y' = 2(C₁ + C₂x)e^2x + (C₂)e^2x.

At x=0: y'(0) = 2C₁ + C₂ = 5 → 4 + C₂ = 5 → C₂ = 1.

Final solution

y(x) = (2 + x)e^2x.

Interpreting Results

The general solution for a second order differential equation with repeated roots includes:

A term with the repeated root raised to the power of x
A term with x multiplied by the repeated root raised to the power of x
Arbitrary constants that are determined by initial conditions

The solution represents a combination of exponential growth/decay and a linear term, which accounts for the repeated root in the characteristic equation.

FAQ

What is a second order differential equation with repeated roots?

It's a special case where the characteristic equation has a double root. The general solution includes an additional term with the natural logarithm function.

How do I know if my equation has repeated roots?

Check if the discriminant (b² - 4ac) of the characteristic equation is zero. If it is, there are repeated roots.

What are the initial conditions used for?

Initial conditions help determine the arbitrary constants in the general solution. They specify the value of the function and its derivative at a particular point.

Can this calculator solve non-linear equations?

No, this calculator is designed for linear, homogeneous second order differential equations with constant coefficients and repeated roots.