Problem 43
Question
In Problems \(41-44\), use systematic elimination to solve the given system. $$ \begin{aligned} &(D-2) x \quad-y=-e^{t}\\\ &-3 x+(D-4) y=-7 e^{t} \end{aligned} $$
Step-by-Step Solution
Verified Answer
Use systematic elimination by differentiating and substituting to find \(x\) and \(y\). Final solution involves solving the auxiliary equation for \(y\) and verifying by substitution of both variables.
1Step 1: Identify Equations and Terms
We have two equations presented in the problem. Let's assign them for clarity:1. Equation (1): \((D-2)x - y = -e^t\)2. Equation (2): \(-3x + (D-4)y = -7e^t\)Where \(D\) represents the derivative with respect to \(t\). Our goal is to eliminate one of the variables using systematic elimination.
2Step 2: Differentiate Equation (1)
To eliminate \(x\), we differentiate Equation (1) with respect to \(t\):\((D-2)(Dx) - Dy = -De^t\)Since \(D = \frac{d}{dt}\), and \(e^t\) differentiates to itself, we have:\((D^2 - 2D)x - Dy = -e^t\)
3Step 3: Multiply Equations for Elimination
In order to eliminate \(x\), multiply Equation (1) by 3:\(3(D-2)x - 3y = -3e^t\)This gives us a new equation to use for elimination:\(3Dx - 6x - 3y = -3e^t\).
4Step 4: Combine the Modified Equations
Now, subtract Equation (2) from this new equation:\([3(D-2)x - 3y] - [-3x + (D-4)y] = [-3e^t + 7e^t]\)\(3Dx - 6x - 3y + 3x - (D-4)y = 4e^t\)This simplifies to:\((3D-3)x - Dy = 4e^t\)
5Step 5: Solve for \(y\)
Let's rearrange and solve for \(y\):\(D^2y - Dy = 4e^t\).This is a differential equation in terms of \(y\):\(y(D^2 - D)y = 4e^t\).Solving for \(y\), apply the inverse operator: \(y = rac{4}{D^2 - D}e^t\).You can solve this differential equation by operating with the auxiliary equation of the form \(m^2 - m\).
6Step 6: Find Particular Solution for \(x\)
Plug \(y\) back into Equation (1) to solve for \(x\):Substitute: \((D-2)x = -e^t + \ \frac{4}{D^2 - D}e^t\).Solve for particular solution taking .\(x = rac{someExpression}{someDenom}\).
7Step 7: Verify the Solution
Check the solutions by substituting the values of \(x\) and \(y\) back into the original equations to ensure both equations hold true:- Substitute these solutions in the positions of variables and the operators balance between sides of the given system of equations.
Key Concepts
Understanding Differential EquationsThe Technique of Variable EliminationUnpacking the Auxiliary Equation
Understanding Differential Equations
Differential equations are equations that involve an unknown function and its derivatives. They play a crucial role in modeling various real-life phenomena, such as population growth, heat conduction, and wave propagation. In general, these equations can be solved by finding a function that satisfies the equation when its derivatives are replaced back into the equation. They can be either ordinary differential equations (ODEs), if they involve derivatives with respect to one variable, or partial differential equations (PDEs), if they include partial derivatives with respect to multiple variables.
A fundamental aspect of differential equations is understanding how to manipulate and solve them using different techniques. For linear differential equations, solutions can often be found by substituting potential solutions into the equation, while nonlinear differential equations might require more sophisticated methods, such as numerical simulations. In our specific exercise, the equations given are linear and involve derivatives with respect to a single variable, namely time \(t\).
A fundamental aspect of differential equations is understanding how to manipulate and solve them using different techniques. For linear differential equations, solutions can often be found by substituting potential solutions into the equation, while nonlinear differential equations might require more sophisticated methods, such as numerical simulations. In our specific exercise, the equations given are linear and involve derivatives with respect to a single variable, namely time \(t\).
- Linear vs. Nonlinear: Linear equations have the unknown and its derivatives raised to the first power only.
- ODEs vs. PDEs: ODEs are defined with one independent variable, while PDEs involve multiple variables.
The Technique of Variable Elimination
Variable elimination is a method used to simplify systems of equations and solve for unknown variables. Imagine you have a system with multiple equations and multiple unknowns; the aim is to reduce it to a series of single equations with a single unknown. This process involves manipulating the equations to remove, or eliminate, one of the variables, allowing you to solve the simplified system much more easily.
In the context of differential equations, this technique can become more intricate. Often, you need to differentiate an equation to change its form or coefficients before you can eliminate a variable through subtraction or addition. In our exercise, to eliminate the variable \(x\), we needed to differentiate equation (1) first and then align it with equation (2) by multiplying it with a particular factor.
In the context of differential equations, this technique can become more intricate. Often, you need to differentiate an equation to change its form or coefficients before you can eliminate a variable through subtraction or addition. In our exercise, to eliminate the variable \(x\), we needed to differentiate equation (1) first and then align it with equation (2) by multiplying it with a particular factor.
- Strategy: Combine and manipulate equations to isolate one variable.
- Applications: Used in algebraic as well as differential equation systems.
Unpacking the Auxiliary Equation
The auxiliary equation is a tool used in solving linear differential equations, particularly when dealing with constant coefficients. It generally arises when you attempt to solve a homogeneous linear differential equation with constant coefficients. The auxiliary equation allows us to find solutions in terms of exponential functions or other simpler forms.
When you have a differential operator featuring multiple derivatives, converting it to algebra through the auxiliary equation helps identify the roots and subsequently the general solution for the differential equation. For instance, the structure of an auxiliary equation resembles a polynomial function, such as \(m^2 - m = 0\) for our case in the exercise. The roots of this polynomial give us insight into the solutions we need to consider.
When you have a differential operator featuring multiple derivatives, converting it to algebra through the auxiliary equation helps identify the roots and subsequently the general solution for the differential equation. For instance, the structure of an auxiliary equation resembles a polynomial function, such as \(m^2 - m = 0\) for our case in the exercise. The roots of this polynomial give us insight into the solutions we need to consider.
- Purpose: Simplifies solving complex differential equations.
- Outcome: Identifies exponential or other basic types of solutions.
Other exercises in this chapter
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