Domain of a Differential Equation

In summary, the particular solution y=f(x) to dy/dx=(1+y)/x given that f(-1)=1 has a domain of x<0 because the discontinuity at x=0 causes the solution to be non-unique. The domain for a particular differential equation is found by considering the initial conditions and ensuring a unique solution for all values of x that satisfy the differential equation.
  • #1
altcmdesc
66
0
Find the particular solution y=f(x) to dy/dx=(1+y)/x given that f(-1)=1 and state its domain.

My answer was: 2|x|-1, x ∈ ℝ/{0}

Apparently, the domain should be x<0

Solving the differential equation was not an issue at all, but I have no idea why the domain is restricted to x<0. I understand that zero cannot be included because dy/dx fails to exist at x=0, but why are positive real numbers not included?

In general, as well, how is the domain of a particular differential equation found that is not obvious (e.g. found by simply looking at dy/dx)
 
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  • #2
With the domain of a "particular" solution for a diff.eq, we mean those x's for which the given intial conditions yield a UNIQUE solution.

In your case, an equally good solution is:

y(x)=2|x|-1, x less than 0, and y(x)=3x-1 for x greater than 0.

The discontinuity at x=0 wreaks havoc on the uniqueness of the solution that normally is guaranteed by specifying a SINGLE initial condition.
 
  • #3
Notice that if you were given exactly the same differential equation with the condition f(1)= 1, the domain would be the set of all positive numbers.
 
  • #4
Ah, I see now. For an initial condition problem, the domain of the solution curve is the set of all x that satisfies the differential equation and contains the initial point. So, since there's a singularity at x=0, only the left half of the curve is the solution. Is my reasoning correct?
 
  • #5
Indeed.

If you were to have a unique solution covering all of R except 0, you'd need TWO initial conditions, one in each of the two disconnected parts of the domain.
 

Related to Domain of a Differential Equation

What is the domain of a differential equation?

The domain of a differential equation refers to the set of all possible values for the independent variable in the equation. It is the range of values for which the equation is valid and can be solved.

Why is the domain of a differential equation important?

The domain of a differential equation is important because it determines the validity of the solution. If the independent variable is outside of the domain, the solution is not considered valid. Additionally, understanding the domain can help with choosing appropriate methods for solving the differential equation.

How can I determine the domain of a differential equation?

The domain of a differential equation can be determined by looking at the given equation and identifying any restrictions on the independent variable. These restrictions may be due to physical or mathematical limitations, such as avoiding division by zero or negative values in a logarithmic function.

What happens if the independent variable is outside of the domain?

If the independent variable is outside of the domain, the solution to the differential equation is not considered valid. This means that the solution does not accurately represent the behavior of the system being modeled by the equation.

Can the domain of a differential equation change?

Yes, the domain of a differential equation can change depending on the context or application. For example, in certain physical systems, the domain may change due to changing conditions or constraints. Additionally, when working with multiple equations, the domain may need to be adjusted to ensure consistency between the equations.

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