Solution Found!
idea. Here is a little exercise in cleverness: Although
Chapter 2, Problem 43E(choose chapter or problem)
Differential equations are sometimes solved by having a clever idea. Here is a little exercise in cleverness: Although the differential equation \(\left(x-\sqrt{x^{2}+y^{2}}\right) \ d x+y \ d y=0\) is not exact, show how the rearrangement \((x d x+y \ d y) / \sqrt{x^{2}+y^{2}}=\ d x\) and the observation \(\frac{1}{2} d\left(x^{2}+y^{2}\right)=x \ d x+y \ d y\) can lead to a solution.
Text Transcription:
(x - sqrt x^2 + y^2) dx + y dy = 0
(x dx + y dy) / sqrt x^2 + y^2 = dx
1/2 d(x^2 + y^2) = x dx + y dy
Questions & Answers
QUESTION:
Differential equations are sometimes solved by having a clever idea. Here is a little exercise in cleverness: Although the differential equation \(\left(x-\sqrt{x^{2}+y^{2}}\right) \ d x+y \ d y=0\) is not exact, show how the rearrangement \((x d x+y \ d y) / \sqrt{x^{2}+y^{2}}=\ d x\) and the observation \(\frac{1}{2} d\left(x^{2}+y^{2}\right)=x \ d x+y \ d y\) can lead to a solution.
Text Transcription:
(x - sqrt x^2 + y^2) dx + y dy = 0
(x dx + y dy) / sqrt x^2 + y^2 = dx
1/2 d(x^2 + y^2) = x dx + y dy
ANSWER:Step 1 of 2
In this problem we have to find the solution of the given differential equation.
Given differential equation is
This equation can be rewritten as
Then
