Which, if any, of the given differential equations is scale invariant? Show how you test each equation, but do not solve them. x+y A. dy da = B. (3xy+y²) dx + (x² + xy) dy = 0
Which, if any, of the given differential equations is scale invariant? Show how you test each equation, but do not solve them. x+y A. dy da = B. (3xy+y²) dx + (x² + xy) dy = 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Title: Scale Invariance of Differential Equations**
**Description:**
This section addresses the scale invariance of differential equations and outlines how to test this property without solving the equations. Two differential equations are presented for examination.
**Equation A:**
\[
\frac{dy}{dx} = \frac{x + y}{x}
\]
This is a first-order differential equation. To test for scale invariance, introduce a scaling factor and check if the equation retains its form.
**Equation B:**
\[
(3xy + y^2) \, dx + (x^2 + xy) \, dy = 0
\]
This is a differential equation in a more complex format, involving both \(dx\) and \(dy\) terms. You can test its scale invariance by replacing \(x\) and \(y\) with scaled variables and determining if the equation simplifies to the original form.
**Note:**
For both equations, apply scaling transformations \(x \rightarrow \lambda x\) and \(y \rightarrow \lambda y\) and analyze if the equations maintain their form under these transformations, indicating scale invariance.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d2ea031-a982-4626-b386-36789c43856b%2Ff8d55772-1ade-4bfb-89f1-0bdf6050953c%2F5xucit_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Scale Invariance of Differential Equations**
**Description:**
This section addresses the scale invariance of differential equations and outlines how to test this property without solving the equations. Two differential equations are presented for examination.
**Equation A:**
\[
\frac{dy}{dx} = \frac{x + y}{x}
\]
This is a first-order differential equation. To test for scale invariance, introduce a scaling factor and check if the equation retains its form.
**Equation B:**
\[
(3xy + y^2) \, dx + (x^2 + xy) \, dy = 0
\]
This is a differential equation in a more complex format, involving both \(dx\) and \(dy\) terms. You can test its scale invariance by replacing \(x\) and \(y\) with scaled variables and determining if the equation simplifies to the original form.
**Note:**
For both equations, apply scaling transformations \(x \rightarrow \lambda x\) and \(y \rightarrow \lambda y\) and analyze if the equations maintain their form under these transformations, indicating scale invariance.
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