Use method of elimination to find the general solution to the system x'-x-y=e¹ ly'-x+y=et
Use method of elimination to find the general solution to the system x'-x-y=e¹ ly'-x+y=et
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Method of Elimination for Solving System of Differential Equations**
**Objective:**
Use the method of elimination to find the general solution to the given system of equations:
\[ \left\{ \begin{array}{l}
x' - x - y = e^t \\
y' - x + y = e^{-t}
\end{array} \right. \]
This problem involves solving a system of linear differential equations using the method of elimination. The equations provided involve derivatives (denoted by \(x'\) and \(y'\)), variables (\(x\) and \(y\)), and exponential functions (\(e^t\) and \(e^{-t}\)).
**Steps to Solve:**
1. **Equation 1:**
\[ x' - x - y = e^t \]
2. **Equation 2:**
\[ y' - x + y = e^{-t} \]
To eliminate one of the variables (either \(x\) or \(y\)), we can manipulate and combine these equations. Here, we aim to eliminate \(y\):
3. **Add Equation 1 and Equation 2:**
\[ (x' - y') - (x - x) - (y + y) = e^t + e^{-t} \]
This simplifies to:
\[ x' + y' - 2y = e^t + e^{-t} \]
4. **Subtraction (Derivative):**
Take the derivative of both equations, as necessary, and continue the elimination by combining these derivatives if needed, until you isolate \(x\) and \(y\).
5. **Finding Solutions:**
Notice the modified equations and repeatedly substitute the derived expressions back into the remaining equations until a solution for \(x\) and \(y\) is found.
6. **General Solution:**
Solve for \(x(t)\) and \(y(t)\) in terms of \(t\). The general solutions will be expressed as a combination of the particular solutions (that fit the non-homogeneous part \(e^t\) and \(e^{-t}\)) and complementary solutions (associated with the homogeneous part of the equations).
**Summary:**
Integrate this process by getting common denominators and combining like terms to reach the general solutions for \(x\) and \(y\). Ensure that](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc59b3e01-f4fe-4ed0-a002-b05c91db4102%2Fc71378ea-94a4-4ce9-a575-feab0fa6d8eb%2F67lmlx_processed.png&w=3840&q=75)
Transcribed Image Text:**Method of Elimination for Solving System of Differential Equations**
**Objective:**
Use the method of elimination to find the general solution to the given system of equations:
\[ \left\{ \begin{array}{l}
x' - x - y = e^t \\
y' - x + y = e^{-t}
\end{array} \right. \]
This problem involves solving a system of linear differential equations using the method of elimination. The equations provided involve derivatives (denoted by \(x'\) and \(y'\)), variables (\(x\) and \(y\)), and exponential functions (\(e^t\) and \(e^{-t}\)).
**Steps to Solve:**
1. **Equation 1:**
\[ x' - x - y = e^t \]
2. **Equation 2:**
\[ y' - x + y = e^{-t} \]
To eliminate one of the variables (either \(x\) or \(y\)), we can manipulate and combine these equations. Here, we aim to eliminate \(y\):
3. **Add Equation 1 and Equation 2:**
\[ (x' - y') - (x - x) - (y + y) = e^t + e^{-t} \]
This simplifies to:
\[ x' + y' - 2y = e^t + e^{-t} \]
4. **Subtraction (Derivative):**
Take the derivative of both equations, as necessary, and continue the elimination by combining these derivatives if needed, until you isolate \(x\) and \(y\).
5. **Finding Solutions:**
Notice the modified equations and repeatedly substitute the derived expressions back into the remaining equations until a solution for \(x\) and \(y\) is found.
6. **General Solution:**
Solve for \(x(t)\) and \(y(t)\) in terms of \(t\). The general solutions will be expressed as a combination of the particular solutions (that fit the non-homogeneous part \(e^t\) and \(e^{-t}\)) and complementary solutions (associated with the homogeneous part of the equations).
**Summary:**
Integrate this process by getting common denominators and combining like terms to reach the general solutions for \(x\) and \(y\). Ensure that
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