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Distinct Real Roots
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Let's solve our first example. We have the equation d²y/dx² + 7 dy/dx + 12y = 0. To start, we need to find the auxiliary equation.
How do we derive the auxiliary equation from this differential equation?
Good question! The auxiliary equation is derived by replacing d²y/dx² with m², dy/dx with m, and y with 1. So it becomes m² + 7m + 12 = 0. Now we can use the quadratic formula.
I see, so we get m = -3 and m = -4, correct?
Exactly! This gives us distinct real roots. Thus, the general solution will be y(x) = C₁e^{-3x} + C₂e^{-4x}.
What does this solution represent in a physical context?
This solution is indicative of exponential decay, commonly found in processes like damped vibrations. Remember, distinct roots indicate a unique behavior in our solutions.
So the key takeaway here is how to identify and interpret the roots of an auxiliary equation for a second-order differential equation. Let's summarize: the auxiliary equation helps derive the general solution, and the nature of roots tells us about the system's behavior.
Repeated Roots
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Now, let's move on to our second exercise. Our equation here is d²y/dx² + 6dy/dx + 9y = 0. Can anyone tell me the form of the auxiliary equation?
It would be m² + 6m + 9 = 0, right?
Yes! Now let's solve it. What do we get?
I think we get a repeated root at m = -3.
Exactly! In this case, since we have a repeated root, how would the general solution look?
It should be y(x) = (C₁ + C₂x)e^{-3x}, since we have to account for the multiplicity.
Correct! This solution implies that the system's response will decay exponentially, but with a linear modification due to repeated roots.
To recap, when we encounter repeated roots, we add an x term to our constants in the general solution. It signifies a different interaction compared to distinct roots.
Complex Roots
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Let's finish with the last exercise involving complex roots. Our equation is d²y/dx² + 16y = 0. What can we do when we see this form?
We can rewrite it as m² + 16 = 0 to find the roots.
Exactly! Can someone tell me what the roots look like?
I think it gives us m = ±4i, which are complex roots.
Correct! With complex roots, how do we form our general solution?
It would be in the form y(x) = e^{0x}(C cos(4x) + C sin(4x)) because alpha is zero in this case.
Right! This form shows oscillatory behavior typical in mechanical vibrations. The solution represents sinusoidal motion, which is crucial in engineering applications.
The take-home message is that complex roots yield oscillatory solutions, highly relevant for studying systems like oscillators or waves.
Introduction & Overview
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Quick Overview
Standard
The section contains three key exercises where the auxiliary equations are derived, solved, and interpreted, illustrating distinct, repeated and complex roots, alongside the general solutions for each case.
Detailed
Detailed Summary
This section focuses on the application of theory from earlier parts of the chapter to solved exercises involving second-order linear homogeneous differential equations. It features three exercises, each demonstrating different scenarios:
1. Distinct Real Roots: The first exercise involves solving a differential equation, presenting an auxiliary equation with distinct roots that lead to an exponential growth solution.
2. Repeated Roots: The second exercise introduces the concept of repeated roots, showcasing how the solution adapts with an additional linear term.
3. Complex Roots: The final exercise demonstrates complex roots, yielding sinusoidal solutions indicative of oscillatory behavior.
Each example is solved methodically, reinforcing the connection between the theory of differential equations and its practical applications in civil engineering.
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Exercise 1: Distinct Real Roots
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Problem: Solve
d2y dy
+7 +12 y=0
dx2 dx
Solution: Auxiliary equation:
m^2 + 7m + 12 = 0
Solving:
−7±√49−48
−7±1
m= = ⇒ m = −3, m = −4
2 2
Hence, the general solution is:
y(x)= C e^(-3x) + C e^(-4x)
Detailed Explanation
In this exercise, we are tasked with solving a second-order linear homogeneous differential equation. The equation given is of the form d²y/dx² + 7 dy/dx + 12y = 0. First, we form the auxiliary equation, which is obtained by substituting y with e^(mx). The auxiliary equation is m² + 7m + 12 = 0. We then solve this quadratic equation using the quadratic formula. The roots are found to be m = -3 and m = -4. With these distinct real roots, the general solution of the differential equation is expressed as y(x) = C₁ e^(-3x) + C₂ e^(-4x), where C₁ and C₂ are arbitrary constants that will be determined by initial or boundary conditions.
Examples & Analogies
Imagine you are trying to model the cooling of a hot cup of coffee. The temperature over time can follow an exponential decay pattern, similar to our solution. Just like the drink cools down faster initially and then slows down, the solution of y(x) mirrors that behavior with distinct terms controlling the cooling at different rates.
Exercise 2: Repeated Roots
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Problem: Solve
d2y dy
+6 +9y=0
dx2 dx
Solution: Auxiliary equation:
m^2 + 6m + 9 = 0 ⇒ m = -3 (repeated)
General solution:
y(x) = (C + Cx)e^(-3x)
Detailed Explanation
This exercise involves a second-order linear homogeneous differential equation where the coefficients allow for repeated roots. After constructing the auxiliary equation, we find that it simplifies to (m + 3)² = 0, yielding the repeated root m = -3. For repeated roots, the general solution takes a different form compared to distinct roots. Here, we express the solution as y(x) = (C₁ + C₂x)e^(-3x). This formulation allows the solution to accommodate the multiplicity of the root, introducing a linear term multiplied by the exponential decay.
Examples & Analogies
Think of this in terms of how a car slows down when applying the brakes. If you press gently, the car slows consistently; however, if you press too hard and keep it pressed, the rate of slowing asymptotically approaches a stop. The repeated roots in our equation represent this consistent reduction in 'speed', showing how the solution adjusts as time goes on.
Exercise 3: Complex Roots
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Problem: Solve
d2y
dx2
−10y=0
Solution: Auxiliary equation:
m^2−10=0 ⇒ m=±√10
General solution:
y(x)=C e^(√10x) + C e^(−√10x)
Detailed Explanation
In this exercise, we address a case of complex roots which aren't immediately obvious from the original differential equation. We manipulate the equation to find the auxiliary form indicating m² = 10. This leads us to m = ±√10, providing us with two complex solutions: one exponentially increasing and the other decreasing. The general solution is expressed in terms of these solutions: y(x) = C₁ e^(√10x) + C₂ e^(−√10x). This representation is essential as it indicates behavior associated with oscillating systems, where one solution grows while the other decays.
Examples & Analogies
Consider a pendulum swinging. When you push it, it oscillates back and forth. The complex roots in our solution represent this oscillatory behavior, where the real part could indicate the ‘stability’ and the imaginary part influences the ‘frequency’ of the swinging motion. The combined exponential terms can depict situations where systems exhibit both growth and decay at the limits of their movement.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Auxiliary Equation: A pivotal equation necessary for finding the characteristic roots of a differential equation.
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General Solution: An expression encapsulating all possible solutions derived from distinct, repeated, or complex roots.
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Distinct Roots: Imply unique solutions yielding exponential growth or decay.
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Repeated Roots: Require an adjustment in the solution due to the multiplicity of roots.
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Complex Roots: Indicate oscillatory behavior in solutions, essential for modeling dynamic systems.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Exercise 1 Solution: y(x) = C e^{-3x} + C e^{-4x} for distinct roots.
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Exercise 2 Solution: y(x) = (C + C x)e^{-3x} for repeated roots.
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Exercise 3 Solution: y(x) = C cos(4x) + C sin(4x) for complex roots.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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Roots can be real or repeat, for solving ODEs, that’s a treat!
📖 Fascinating Stories
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Imagine an engineer named Otto, who found variable roots: some made beams strengthen while others made them wobble harmoniously.
🧠 Other Memory Gems
-
Remember DRR (Distinct Real Roots), RR (Repeated Roots), CR (Complex Roots) for each type.
🎯 Super Acronyms
Use R, R, C for Roots
- Real (distinct)
- Real (repeated)
- Complex.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Auxiliary Equation
Definition:
An equation derived from a differential equation to find the characteristic roots.
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Term: General Solution
Definition:
The comprehensive expression representing the complete set of solutions to a differential equation.
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Term: Complex Roots
Definition:
Roots of the auxiliary equation that are not real, typically expressed in the form of a+bi.
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Term: Distinct Real Roots
Definition:
Different real values obtained from solving the auxiliary equation, leading to an exponentially decaying solution.
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Term: Repeated Roots
Definition:
Two identical values from the auxiliary equation, requiring adjustments in the general solution.