Easy Seperable Differential Equation Domain Questions

In summary, the author of the Boyce and Diprima Elementary DE book notes that the solution to a certain equation is defined on the interval (-2,3) only. However, the solution can also be defined on the intervals (-inf,-2) and (3,inf), but these solutions cannot be connected to the one between -2 and 3. This is not a mistake in the book, as the solution satisfying the given boundary condition is only valid between -2 and 3.
  • #1
danerape
32
0
This is a problem from the Boyce and Diprima Elementary DE book. I solved this equation
and got the correct solution. However, the author notes that the solution is defined on the
interval (-2,3) only. I have found it to be defined (-inf,-2)u(-2,3)u(3,inf), or EVERYWHERE BUT x=-2, x=3. Is this a mistake in the book or am I missing something?


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Thanks Alot,

Dane
 

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  • #2
Yes, the book is correct. There exist solutions for x< -2 and x> 3 but they cannot be "connected" to the solution between -2 and 3. The solution satisfying the given boundary condition is valid only between -2 and 3.
 

FAQ: Easy Seperable Differential Equation Domain Questions

What is a separable differential equation?

A separable differential equation is a type of differential equation that can be manipulated into a form where the dependent and independent variables are separated on opposite sides of the equation. This allows for the integration of each side separately, making it easier to solve the equation.

What is the domain of a differential equation?

The domain of a differential equation refers to the set of values that the independent variable can take on. It is important to determine the domain before solving a differential equation to ensure that the solution is valid for all values within the domain.

How do I identify a separable differential equation?

A separable differential equation is typically in the form of dy/dx = f(x)g(y), where f(x) and g(y) are functions of the independent and dependent variables, respectively. Additionally, the equation can be manipulated into this form by separating the variables on opposite sides of the equation.

What are the steps for solving a separable differential equation?

The steps for solving a separable differential equation are as follows:
1. Identify the independent and dependent variables
2. Rearrange the equation into the form dy/dx = f(x)g(y)
3. Integrate both sides of the equation with respect to the appropriate variables
4. Solve for the dependent variable to find the general solution
5. Use initial conditions to find the particular solution, if needed.

Can any differential equation be solved using separation of variables?

No, not all differential equations can be solved using separation of variables. This method can only be used for equations that are in the form dy/dx = f(x)g(y). Other types of differential equations may require different methods to solve them.

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