Show that the given function is a solution of the differential equation. dy dx dy dx || +9y = e-9x . y = xe -9x -9x + 8e dy Substituting and y into the original equation gives dx +9(xe -9x + 8e-9x) = e-⁹x. The solution checks.
Show that the given function is a solution of the differential equation. dy dx dy dx || +9y = e-9x . y = xe -9x -9x + 8e dy Substituting and y into the original equation gives dx +9(xe -9x + 8e-9x) = e-⁹x. The solution checks.
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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Show that the given function is a solution of the
*** Please see attachment
![**Differential Equation Verification**
To determine if the given function is a solution of the differential equation:
\[
\frac{dy}{dx} + 9y = e^{-9x}, \quad y = xe^{-9x} + 8e^{-9x}
\]
**Step 1: Differentiate \( y \)**
\[
\frac{dy}{dx} = \text{[Compute the derivative of \( y = xe^{-9x} + 8e^{-9x} \) here]}
\]
**Step 2: Substitute \(\frac{dy}{dx}\) and \( y \) into the original equation**
\[
\text{Substituting } \frac{dy}{dx} \text{ and } y \text{ into the original equation gives:}
\]
\[
\quad [\frac{dy}{dx} \text{ (calculated value)}] + 9(xe^{-9x} + 8e^{-9x}) = e^{-9x}
\]
The solution checks out as the left side equals the right side.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50b7307d-f147-4c39-812e-01e48cbfa721%2Fe404aa4a-3a7d-43b3-89a7-8d5007e7c4e7%2Fa8ymt0k_processed.png&w=3840&q=75)
Transcribed Image Text:**Differential Equation Verification**
To determine if the given function is a solution of the differential equation:
\[
\frac{dy}{dx} + 9y = e^{-9x}, \quad y = xe^{-9x} + 8e^{-9x}
\]
**Step 1: Differentiate \( y \)**
\[
\frac{dy}{dx} = \text{[Compute the derivative of \( y = xe^{-9x} + 8e^{-9x} \) here]}
\]
**Step 2: Substitute \(\frac{dy}{dx}\) and \( y \) into the original equation**
\[
\text{Substituting } \frac{dy}{dx} \text{ and } y \text{ into the original equation gives:}
\]
\[
\quad [\frac{dy}{dx} \text{ (calculated value)}] + 9(xe^{-9x} + 8e^{-9x}) = e^{-9x}
\]
The solution checks out as the left side equals the right side.
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