The loss of y = 0 as a solution

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In summary, when solving a differential equation and dividing by a variable, it is important to check the case when that variable is equal to 0, as it may be a solution that is lost in the final solution. This can be done by plugging in y = 0 into the original differential equation and verifying that both sides are equal. This is known as the "trivial equilibrium solution" and should not be ignored.
  • #1
djeitnstine
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When solving a Diff. Eq. how do we know that y=0 is another solution lost when we solved it?
 
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  • #2
You don't know how to check if y = 0 is a solution of a DE?
 
  • #3
Whatever you divided by is potentially lost as a (when set to zero)solution of your differential equation. So for example, y2 = y*y' you divide by y to get y=y' and you know the solution of that, but since you divided both sides by y, you need to check the case when y=0, since you can't divide by y when y=0
 
  • #4
Thank you for the replies. For NoMoreExams an example here: I did an ODE [tex]\frac{dy}{dx}-y=e^{2x}y^{3}[/tex]

and solved it using bournulli's method to get

[tex]y=\sqrt{\frac{2}{-e^{2x} + Ce^{-2x}}}[/tex]

My professor said y = 0 was lost in that solution.

So I place y = 0 in the original diff eq. to get

[tex]\frac{dy}{dx}-0=e^{2x}0^{3}[/tex]

which is

[tex]\frac{dy}{dx}=0[/tex]

which seems wierd
 
  • #5
Remember how you check solutions, LHS = RHS

Since

[tex] y = 0 \Rightarrow \frac{dy}{dx} = 0 [/tex]

So let's look at LHS:

[tex] \frac{dy}{dx} - y = 0 - 0 = 0 [/tex]

Now let's look at RHS:

[tex] e^{2x}y^{3} = e^{2x}0^{3} = 0 [/tex]

Since [tex] 0 = 0 [/tex], we have shown that [tex] y = 0 [/tex] is a solution to our DE.
 
  • #6
NoMoreExams said:
[tex] y = 0 \Rightarrow \frac{dy}{dx} = 0 [/tex]

Wow, I can't believe I completely missed that...thanks.
 
  • #7
Never ignore the trivial equilibrium soln.
They're always the hardest, since you're never really "looking hard" for them.
 

Related to The loss of y = 0 as a solution

1. What does "loss of y = 0 as a solution" mean?

The loss of y = 0 as a solution refers to a situation in which the variable y can no longer be equal to 0 in a given equation or problem. This could be due to a change in the parameters or constraints of the equation, making y = 0 an invalid solution.

2. Why is the loss of y = 0 as a solution significant?

The loss of y = 0 as a solution can have significant implications in scientific research and problem solving. It could indicate a change in the underlying system or model being studied, and may require further investigation or adjustments to the equations being used.

3. How does the loss of y = 0 as a solution affect the overall solution of a problem?

The loss of y = 0 as a solution can alter the overall solution of a problem, as it eliminates one possible solution and may require finding a new solution or redefining the problem. It may also impact the accuracy and validity of the solution.

4. What factors can contribute to the loss of y = 0 as a solution?

There are various factors that can lead to the loss of y = 0 as a solution, including changes in the parameters or constraints of the equation, errors in data or calculations, or limitations in the model being used.

5. How can the loss of y = 0 as a solution be addressed in scientific research?

To address the loss of y = 0 as a solution, scientists may need to revisit their assumptions, data, and equations to identify any discrepancies or errors that may have led to this loss. They may also need to modify their models or conduct further experiments to better understand the underlying system.

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