3.3.2 - Case 2: Repeated Real Roots (D =0)
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Introduction to Repeated Roots
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Today we’ll explore what happens when we have repeated real roots in our equations. Can anyone remind me what we mean by a repeated root in the context of characteristic equations?
I think it’s when we have the same root occurring more than once, right?
Exactly! So when our discriminant D equals zero, we encounter this situation. What would be the general solution to such equations?
Is it something like y(x) = (C_1 + C_2 x)e^{rx}?
Right again! We introduce that polynomial term to handle the repeated root. It’s crucial to remember this adjustment. Now, let’s look at an example to clarify.
Deriving the General Solution
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Let’s examine the equation y'' - 4y' + 4y = 0. What do we do first to find the roots?
We need to set up the characteristic equation, right? So that would be r^2 - 4r + 4 = 0.
Correct! Now, how do we solve that?
We can factor it as (r - 2)(r - 2) = 0, which gives us r = 2, a repeated root.
Exactly! So, can someone tell me the general solution now?
It’s y(x) = (C_1 + C_2 x)e^{2x}.
Perfect! It demonstrates how our equation transforms with repeated roots.
Applying Initial Conditions
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Now that we have our general solution, y(x) = (C_1 + C_2 x)e^{2x}, how do we find the arbitrary constants?
We use initial conditions! Like y(0) and y'(0).
Correct! Suppose we have y(0) = 2 and y'(0) = -1. How do we apply those?
For y(0) = 2, we’ll substitute x = 0 into the general solution.
And what does that give us?
It means C_1 should equal 2 since e^{0} = 1.
Exactly! Then for y'(0), we derive the expression and find the second constant.
Significance in Engineering Applications
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Can anyone explain why understanding repeated roots is important in civil engineering?
It likely relates to stability in structures during vibrations!
Spot on! The equations often represent critical points in design and analysis, particularly for cantilever beams or structures prone to oscillation.
So, properly solving these equations can help us predict how structures behave?
Exactly! Accurate solutions are essential for safety and performance. A thorough understanding is crucial.
Introduction & Overview
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Quick Overview
Standard
In this section, we explore the scenario of repeated real roots in second-order homogeneous linear differential equations, where the discriminant equals zero. The general solution is presented, along with a practical example and methods to find constants using initial conditions.
Detailed
Case 2: Repeated Real Roots (D = 0)
In second-order homogeneous linear differential equations with constant coefficients, the nature of the roots derived from the characteristic equation plays a crucial role in determining the general solution. When the discriminant of the characteristic equation is zero (D = 0), it indicates the presence of repeated real roots. This situation necessitates a modified expression for the general solution to account for the multiplicity of the roots.
Characteristics of Repeated Roots
When the roots of the characteristic equation,
$$ ar^2 + br + c = 0 $$
are both equal (let's denote them as $r$), the general solution can be expressed as:
$$ y(x) = (C_1 + C_2 x)e^{rx} $$
where $C_1$ and $C_2$ are arbitrary constants determined by specific initial or boundary conditions.
Example
For instance, given the differential equation:
$$ y'' - 4y' + 4y = 0 $$
we can derive the characteristic equation:
$$ r^2 - 4r + 4 = 0 $$
with a repeated root of $r = 2$. Thus, the general solution is:
$$ y(x) = (C_1 + C_2 x)e^{2x} $$
This extended solution allows for the polynomial term multiplied by the exponential function, accommodating the repeated root scenario effectively. In subsequent sections, we will discuss the practical implications and applications of these solutions in engineering problems.
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Key Concepts
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Repeated Roots: Indicate a unique form in the general solution.
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Homogeneous Equations: Right-hand side is zero, representing a balance.
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General Solution: Functional form that includes arbitrary constants.
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Characteristic Equation: Helps in finding roots that determine the general solution.
Examples & Applications
For y'' - 4y' + 4y = 0, the general solution is y(x) = (C_1 + C_2 x)e^{2x}.
Applying initial conditions of y(0) = 2 and y'(0) = -1 can help find C_1 and C_2.
Memory Aids
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Rhymes
Repeated roots with zero D, leads to solutions that are C's plus x, you see!
Stories
Imagine a builder checking beams. He finds they're stable when roots aren't just dreams. When roots are paired, his designs take flight; In safety's embrace, they stand upright!
Memory Tools
RAPID: Repeated roots Add Polynomial, Important Derivation.
Acronyms
RACE
Roots Are Common with Exponential.
Flash Cards
Glossary
- Homogeneous Equation
A differential equation where the right-hand side is equal to zero.
- Characteristic Equation
The polynomial derived from a differential equation that helps in finding the roots determining the behavior of the solution.
- Repeated Roots
Roots of a polynomial that appear more than once, indicating a unique form in the general solution of a differential equation.
- General Solution
A solution that contains all possible solutions of the differential equation, often expressed with arbitrary constants.
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