Examining Solutions of Non-Linear DEs

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Homework Help Overview

The discussion revolves around the properties of solutions to a non-linear second-order differential equation (DE) given specific solutions y1 = x^2 and y2 = 1. Participants are examining whether transformations of these solutions, specifically -y1 and the sum y1 + y2, also qualify as solutions.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are questioning the validity of -y1 and y1 + y2 as solutions to the DE. One participant attempts to substitute -y1 into the DE to check for consistency, while another confirms the approach for y1 + y2.

Discussion Status

The discussion is active, with participants exploring the implications of their substitutions. One participant has concluded that -y1 does not satisfy the DE based on their calculations, while another has affirmed the correctness of checking y1 + y2 similarly.

Contextual Notes

There is an acknowledgment of the non-linear nature of the DE, which complicates the addition of solutions. Participants are also reflecting on their thought processes, indicating a level of uncertainty in their reasoning.

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Homework Statement



Given y1=x^2 and y2=1 are two solutions of the DE
y"=(x/y)y'

Are the functions -y1 and y1 and y2 also the solutions of the equation?If not why?

Homework Equations


The Attempt at a Solution



I cannot see how to proceed.However,I can see that it is a non-linear DE of
2nd degree where we cannot simply sum the solutions to have another solution...
That might explain the case that y1+y2 is not a solution...

what about the first case,i.e -y1?
 
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Do you end up with an identity when you substitute -y1, (-y1)' and (-y1)'' the DE?
 
OK what I am getting is a wrong thing:-2=2!
So,this shows -y1 cannot be a solution.

Perhaps I was thinking in unnecessary complicated manner.

What about the 2nd part? that is y1+y2?
Am I correct there?
 
Yes you are. You can check by doing what you did for 1.
 

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