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Understanding Simultaneous Linear Differential Equations
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Today we are going to explore how we can use the Laplace Transform to solve simultaneous linear differential equations. Who can tell me what simultaneous equations are?
They are equations that must be solved together because they share variables.
Exactly! In terms of differential equations, this means we have equations involving the same functions of time, x(t) and y(t). Can someone give me an example of where we might see these equations in real life?
Maybe in electrical circuits where currents depend on each other?
Good example! We apply Laplace Transform here because it simplifies our task by transforming these differential equations into algebraic equations. Remember: 'Laplace is Great for Algebraic Fate!' How does that sound?
Sounds catchy! It helps remember the purpose of Laplace Transforms!
Let's move on and see how to apply it step by step.
Applying the Laplace Transform
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We start by taking the Laplace Transform of both equations. What do we do with initial conditions when we perform this step?
We include them in our equations!
Correct! Let's represent our equations as the Laplace Transform equations. For example, if we have dx/dt = 3x + 4y, how does it transform?
It becomes sX(s) - x(0) = 3X(s) + 4Y(s).
Great! We gather them into a system of algebraic equations. Can anyone tell me why this setup is so beneficial?
Because algebraic equations are easier to solve than differential ones!
Exactly! Remember that. Now, let's solve for one variable in terms of another.
Solving the Algebraic Equations
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Now that we have our equations, we want to solve them simultaneously. What strategies can we use for solving these?
Substitution or elimination could work!
Absolutely! Letβs say we express Y in terms of X or vice versa. How do we find one variable using substitution?
We replace Y in one equation with the expression from another equation.
Right on! After substitution, we can derive expressions for both X(s) and Y(s). Strongly rememberβ'Substitute to Solve and Evolve!' Now, letβs dive into Inverse Laplace Transforms.
Inverse Laplace Transform
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The final step is to apply the Inverse Laplace Transform. Can someone remind me what this step does?
It takes us back from the s-domain to the time domain!
Correct! And why is this important?
Because we want to see how the functions behave over time!
Exactly! By using standard transforms, we convert X(s) and Y(s) back. Great motto: 'Transform to See Functions Flea!' Finally, who can write the final expressions for x(t) and y(t)?
x(t) = e^{3t}cos(4t) and y(t) = -e^{3t}sin(4t).
Fantastic! You have learned a powerful technique for solving interconnected systems.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
This section presents a specific example where a system of simultaneous linear differential equations is solved using the Laplace Transform technique. The steps include taking Laplace Transforms, rearranging the equations, and applying the Inverse Laplace Transform to derive the solutions.
Detailed
Detailed Summary
In this section, we solve a set of simultaneous linear differential equations using Laplace Transforms. The equations capture the dynamics of interconnected systems, prevalent in engineering scenarios. The main steps in solving such equations involve:
- Taking the Laplace Transform of both equations while accounting for initial conditions, yielding algebraic forms.
- Rearranging these forms into a system of algebraic equations, allowing for manipulations to isolate the desired function.
- Solving simultaneously for both unknown functions, generally using methods like substitution or elimination to establish a relationship between the functions.
- Applying the Inverse Laplace Transform to revert to the time domain, retrieving the functions that describe the system's behavior over time.
The section concludes with a practical worked example where the system's behavior is computed, illustrating each step of the process, from the initial equations through to the final time-domain solutions.
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Problem Statement
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Problem:
Solve the system using Laplace Transform:
dx/dt = 3x + 4y,
dy/dt = -4x + 3y,
x(0) = 1, y(0) = 0
Detailed Explanation
In this section, we are presented with a problem that involves solving a system of simultaneous linear differential equations using the Laplace Transform method. The equations given are:
1. dx/dt = 3x + 4y
2. dy/dt = -4x + 3y
Additionally, we are provided with initial conditions: x(0) = 1 and y(0) = 0. These initial conditions will be crucial when applying the Laplace Transform because they allow us to find the specific solution to our equations.
Examples & Analogies
Imagine you're trying to figure out how two interconnected water tanks behave over time. One tank fills up faster based on the levels in both tanks, while another tank drains water based on the water level in the first tank. The equations given are similar to the rules that govern how the water levels change, and the initial conditions tell us how much water is in each tank at the start.
Step 1: Take Laplace Transforms
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Step 1: Take Laplace Transforms
L{dx/dt} = sX(s) - x(0), L{dy/dt} = sY(s) - y(0)
So the equations become:
sX(s) - 1 = 3X(s) + 4Y(s)
sY(s) - 0 = -4X(s) + 3Y(s)
Detailed Explanation
In this step, we apply the Laplace Transform to both of the differential equations we formulated. By doing so, we convert the differential equations into algebraic equations in the s-domain. Applying the Laplace Transform involves using the property that transforms the derivative of a function into a function in the s-domain, adjusted for initial conditions:
- For dx/dt, we have L{dx/dt} = sX(s) - x(0)
- For dy/dt, we have L{dy/dt} = sY(s) - y(0)
This yields two new equations we can work with.
Examples & Analogies
Think of the Laplace Transform as a way to map a changing situation, like a fluctuating water level in our tanks, into a simpler, static snapshot. Instead of tracking the chaos of changing water levels over time, we're reorganizing things so we can analyze them more easily while keeping track of the starting amounts.
Step 2: Rearranging the Equations
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Step 2: Rearranging the Equations
sX(s) - 3X(s) - 4Y(s) = 1 β (s - 3)X(s) - 4Y(s) = 1 (Equation 1)
sY(s) + 4X(s) - 3Y(s) = 0 β 4X(s) + (s - 3)Y(s) = 0 (Equation 2)
Detailed Explanation
Next, we rearrange the transformed equations to isolate terms involving X(s) and Y(s). For our first equation, after substituting in x(0) = 1, we isolate the term for X(s). In the second equation, we isolate Y(s). This gives us two algebraic equations (Equation 1 and Equation 2) that can be solved simultaneously.
Examples & Analogies
Consider a recipe that gives different ingredients needed to prepare a dish. By rearranging the recipe and isolating certain values, you can determine how much of each ingredient you require. Here, we rearranged our equations similar to tweaking a recipe to make it easier to understand how much of each 'variable' we need.
Step 3: Solve Simultaneously
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Step 3: Solve Simultaneously
From equation (2):
(s - 3)
4X(s) = - (s - 3)Y(s) β X(s) = - Y(s) / 4
Substituting into equation (1):
(s - 3) [-((s - 3)/4)Y(s)] - 4Y(s) = 1
Detailed Explanation
Now, we need to solve the two equations simultaneously. From Equation 2, we expressed X(s) in terms of Y(s). By substituting this expression for X(s) back into Equation 1, we can reduce the equation to just Y(s). This manipulation helps us find a specific form for Y(s) before we evaluate back to find X(s).
Examples & Analogies
Think about trying to balance ingredients in a dish. If you know the ratio of one ingredient to another, you can substitute the known quantities to find out how much of another ingredient you should use. Here, we are substituting Y(s) value to simplify and find X(s) similarly.
Step 4: Take Inverse Laplace Transforms
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Step 4: Take Inverse Laplace Transforms
Using standard transforms:
L^{-1}{(s - a)/((s - a)^2 + b^2)} = e^{at}cos(bt)
L^{-1}{b/((s - a)^2 + b^2)} = e^{at}sin(bt)
So:
X(s) = (s - 3)/((s - 3)^2 + 4^2) β x(t) = e^{3t}cos(4t)
Y(s) = -4/((s - 3)^2 + 4^2) β y(t) = -e^{3t}sin(4t)
Detailed Explanation
After finding X(s) and Y(s), we now take the Inverse Laplace Transform to convert these algebraic expressions back into functions of time, x(t) and y(t). We've utilized known inverse transforms to achieve this, leading to our final solutions for x(t) and y(t). The transforming process allows us to interpret the outcome in real-world time-dependent behaviors.
Examples & Analogies
Imagine flipping a pancake β the batter transforms from a liquid state into a solid form as it cooks. Similarly, while we worked with equations in the s-domain, we now transform them back into meaningful time functions, which allows us to see how the system behaves over time.
Final Answer
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Final Answer:
x(t) = e^{3t}cos(4t),
y(t) = -e^{3t}sin(4t)
Detailed Explanation
The final step provides the solution to the original system of differential equations. We have successfully found that x(t) = e^{3t}cos(4t) and y(t) = -e^{3t}sin(4t). This step demonstrates the effectiveness of the Laplace Transform technique in solving complex systems and showcases how we can express interactions of x and y in the time domain after going through a systematic approach.
Examples & Analogies
Think about predicting the behavior of a pendulum as it swings over time. After going through the whole calculation and transformation, we now have precise predictions (x(t) and y(t)) for how the pendulum would behave at any moment in time, giving us valuable insights into the dynamics of the system.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Laplace Transform: A technique for converting differential equations into algebraic equations.
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Simultaneous Linear Differential Equations: Equations that need to be solved together due to shared variables.
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Inverse Laplace Transform: A method to convert functions from the frequency domain back to time domain.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Solving equations in electrical circuits where multiple currents influence one another through differential relationships.
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Modeling mechanical systems with interconnected parts that influence each other's behavior.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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When systems are tangled, and variables are near,
Use Laplace all the way, your solution is clear!
π Fascinating Stories
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Imagine two rivers joining into a lake. Each river's flow affects the lake's level. Similarly, in simultaneous differential equations, each variable influences the other!
π§ Other Memory Gems
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Remember 'L.A.S.T.' for solving: Laplace, Algebraic Equations, Solve, Transform back to Time!
π― Super Acronyms
S.E.A. - Simultaneous Equations Algebra - for solving systems with Laplace!
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Laplace Transform
Definition:
A mathematical operation that transforms a function of time into a function of a complex variable s, used for solving differential equations.
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Term: Simultaneous Linear Differential Equations
Definition:
A set of differential equations involving multiple variables that must be solved simultaneously.
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Term: Inverse Laplace Transform
Definition:
A process of converting a function in the s-domain back to the time domain.