Examining Solutions of Non-Linear DEs

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In summary, the conversation discusses whether the functions -y1 and y1+y2 are solutions of the differential equation y"=(x/y)y'. It is concluded that -y1 is not a solution, while y1+y2 is a valid solution. The reasoning behind this is that the given DE is non-linear and the solutions cannot simply be summed to obtain another solution. The conversation also mentions that this may have been considered in a complicated manner, but ultimately the conclusion is that y1+y2 is indeed a solution.
  • #1
neelakash
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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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  • #2
Do you end up with an identity when you substitute -y1, (-y1)' and (-y1)'' the DE?
 
  • #3
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?
 
  • #4
Yes you are. You can check by doing what you did for 1.
 

FAQ: Examining Solutions of Non-Linear DEs

What is a non-linear differential equation?

A non-linear differential equation is a mathematical equation that involves derivatives of a function where the dependent variable is raised to a power or multiplied by itself, making it a non-linear function. This means that the rate of change of the function is not proportional to the function itself.

How are non-linear differential equations solved?

Non-linear differential equations can be solved using various methods, such as substitution, separation of variables, or the use of numerical approximations. These methods often involve transforming the non-linear equation into a simpler form that can be solved using known techniques.

What are the applications of non-linear differential equations?

Non-linear differential equations are used in many fields, including physics, engineering, economics, and biology. They are particularly useful in modeling complex systems and phenomena that cannot be accurately represented by linear equations.

How do non-linear differential equations differ from linear differential equations?

In linear differential equations, the dependent variable and its derivatives appear only in the first degree, while in non-linear differential equations, they can appear in higher degrees. This results in different behaviors and solutions for the two types of equations.

What are the challenges in solving non-linear differential equations?

Non-linear differential equations can be more difficult to solve than linear equations due to their complex behaviors and lack of general solutions. They often require approximate methods and computer simulations to find solutions. Additionally, not all non-linear equations have closed-form solutions, making it challenging to find an exact solution.

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