Applying auxiliary equation to find the general solution. y "- 7y"-18y = 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Applying Auxiliary Equation to Find the General Solution**
The given differential equation is:
\[ y'''' - 7y'' - 18y = 0 \]
This equation is a fourth-order linear homogeneous differential equation. The task involves applying an auxiliary equation, often used to find the general solution for such equations.
In the auxiliary equation method, you assume a solution of the form \( y = e^{rt} \), where \( r \) is a constant and \( t \) is a variable representing time or another continuous quantity. Substituting this expression into the differential equation allows you to derive a characteristic equation in terms of \( r \). Solving this characteristic equation provides the values of \( r \), which are used to construct the general solution of the original differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c1e8b90-227b-4a9c-b073-26e8337ecb0b%2Ff7559e2a-96ff-43c3-827f-0edf1a1701ee%2Fpxehlth_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Applying Auxiliary Equation to Find the General Solution**
The given differential equation is:
\[ y'''' - 7y'' - 18y = 0 \]
This equation is a fourth-order linear homogeneous differential equation. The task involves applying an auxiliary equation, often used to find the general solution for such equations.
In the auxiliary equation method, you assume a solution of the form \( y = e^{rt} \), where \( r \) is a constant and \( t \) is a variable representing time or another continuous quantity. Substituting this expression into the differential equation allows you to derive a characteristic equation in terms of \( r \). Solving this characteristic equation provides the values of \( r \), which are used to construct the general solution of the original differential equation.
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