When is a Constant a Solution to a Differential Equation

In summary: Therefore, when one is solving a differential equation and using division one must check what happens if the term is equal to zero, and whether it leads to a singular solution.
  • #1
Drakkith
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I've run across several instances while doing homework where a question will have two solutions. One will be an equation, and the 2nd will be a constant (usually zero). I can't figure out why this constant is a solution though.

For example, take the following differential equation: $$(x^2-3y^2)dx+(2xy)dy=0$$

The solutions to this differential equation turn out to be $$y=±\sqrt{x^2+Cx^3}$$
and$$x=0$$

Why is ##x=0## a solution? The only thing I can think of, and this is mostly a guess, is that plugging ##x=0## into the original equations yields $$(-3y^2)dx=0$$
Is it that ##y## is taken to be a constant here and since there's no ##x## variable there is no change?
 
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  • #2
If x=0 it does not matter what y is because both x and dx are zero and so both terms contain factors that are zero.
 
  • #3
Orodruin said:
because both x and dx are zero

Why is ##dx## zero?
 
  • #4
Because x is constant.
 
  • #5
I'm assuming that's not true for, say, ##x=1##. Is that because the ##dy## term isn't zero then?
 
  • #6
Drakkith said:
I'm assuming that's not true for, say, ##x=1##. Is that because the ##dy## term isn't zero then?
##dx## would still be zero if ##x## was constantly equal to one. Indeed, the ##dy## term is not zero then.
 
  • #7
Alright. Thanks Oro.
 
  • #8
When you solved the differential equation you had to divide both sides by x, since x ≠ 0, x=0 will be omitted from your general solution by default. Remember to write your restrictions, because it's impossible to get singular solutions by varying the parameters of your general solution as far as I know.
 
  • #9
VuIcan said:
When you solved the differential equation you had to divide both sides by x, since x ≠ 0, x=0 will be omitted from your general solution by default. Remember to write your restrictions, because it's impossible to get singular solutions by varying the parameters of your general solution as far as I know.

What restrictions? X=0 was not omitted from the general solution as far as the book is concerned. That was the point of my question.
 
  • #10
Drakkith said:
What restrictions? X=0 was not omitted from the general solution as far as the book is concerned. That was the point of my question.
I misunderstood your question, my apologies.
 
  • #11
VuIcan said:
I misunderstood your question, my apologies.

Well, looking back at the question, I now have another question. The equation can easily be rearranged into ##\frac{dy}{dx}=\frac{3y^2-x^2}{2xy}##
In this form you certainly have a problem when ##x=0##, right?
 
  • #12
Drakkith said:
Well, looking back at the question, I now have another question. The equation can easily be rearranged into ##\frac{dy}{dx}=\frac{3y^2-x^2}{2xy}##
In this form you certainly have a problem when ##x=0##, right?

I'm self-taught on this topic, so take what I say with a grain of salt. But here's what I was referring to:

https://en.wikipedia.org/wiki/Singular_solution

"Therefore, when one is solving a differential equation and using division one must check what happens if the term is equal to zero, and whether it leads to a singular solution."

This is why I urged you to write your restrictions. I assumed you were wondering why your textbook had constant values as solutions. It's because those constant values are omitted on occasion when you perform certain operations(division in this case) from the domains of all the particular solutions obtainable via varying the constant in your general solution.

Again, I'm self taught (still learning in fact) so take what I say with a grain of salt and please do correct me if I'm mistaken. Thanks in advance.
 

Related to When is a Constant a Solution to a Differential Equation

1. What is a constant solution in a differential equation?

A constant solution in a differential equation is a solution in which the dependent variable does not vary with respect to the independent variable. In other words, the value of the dependent variable remains constant throughout the entire domain of the independent variable.

2. How do you determine if a constant is a solution to a differential equation?

To determine if a constant is a solution to a differential equation, you can substitute the constant value into the differential equation and see if it satisfies the equation. If the constant value makes the equation true, then it is a solution to the differential equation.

3. What is the significance of a constant solution in a differential equation?

A constant solution in a differential equation can represent a stable equilibrium point in a system. It can also indicate a situation where the rate of change of the dependent variable is equal to zero, meaning that there is no change over time.

4. Can a constant be the only solution to a differential equation?

Yes, a constant can be the only solution to a differential equation. This is typically the case when the differential equation is a first-order linear equation with no independent variable term. In this case, the solution will be in the form of y = C, where C is a constant.

5. How does the initial condition affect the constant solution to a differential equation?

The initial condition can determine the value of the constant in a constant solution to a differential equation. It acts as a boundary condition that must be satisfied by the solution. If the initial condition is not specified, then the constant can take on any value and the constant solution will still be a valid solution to the differential equation.

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