For a partial differential equation, in a function φ (x, y) and two variables x, y, what is the form obtained after separation of variables is applied?a)Φ (x, y) = X(x) + Y(y)b)Φ (x, y) = X(x) - Y(y)c)Φ (x, y) = X(x) / Y(y)d)Φ (x, y) = X(x)Y(y)Correct answer is option 'D'. Can you explain this answer?

Question

For a partial differential equation, in a function &phi; (x, y) and two variables x, y, what is the form obtained after separation of variables is applied?a)&Phi; (x, y) = X(x) + Y(y)b)&Phi; (x, y) = X(x) - Y(y)c)&Phi; (x, y) = X(x) / Y(y)d)&Phi; (x, y) = X(x)Y(y)Correct answer is option 'D'. Can you explain this answer?

Answer

The method of separation of variables relies upon the assumption that a function of the form,
Φ (x, y) = X(x)Y(y)
will be a solution to a linear homogeneous partial differential equation in x and y. This is called a product solution and provided the boundary conditions are also linear and homogeneous this will also satisfy the boundary conditions.

Answer

The method of separation of variables relies upon the assumption that a function of the form,
Φ (x, y) = X(x)Y(y)
will be a solution to a linear homogeneous partial differential equation in x and y. This is called a product solution and provided the boundary conditions are also linear and homogeneous this will also satisfy the boundary conditions.

Answer

Separation of Variables in a Partial Differential Equation
Separation of variables is a common method used to solve partial differential equations. When applying separation of variables to a partial differential equation in a function φ (x, y) with two variables x and y, the form obtained is typically of the following type:

The Form Obtained
- Φ (x, y) = X(x)Y(y)
This form arises when we assume that the solution to the partial differential equation can be expressed as the product of two functions, each of which only depends on one of the variables. This allows us to separate the variables and simplify the equation into two ordinary differential equations, one for each function.
By separating the variables in this way, we are able to reduce the partial differential equation into simpler ordinary differential equations that can be solved independently. This technique is particularly useful for solving certain types of partial differential equations, such as those that are separable.
Therefore, after applying separation of variables to a partial differential equation in a function φ (x, y) with two variables x and y, the form obtained is typically Φ (x, y) = X(x)Y(y).

Similar Mathematics Doubts

Consider the differential equation ( x + y + 1) dx + (2x + 2y + 1) dy = 0. Which of the following statements is true?

Consider the differential equation (x + y + 1) dx + (2x + 2y + 1)dy = 0. Which of the following statement is true?

Consider the differential equationwhich of the following statements istrue ?

What is the reason behind the non-existence of any real function which satisfies the differential equation, (y’)2 + 1 = 0?

What is the reason behind the non-existence of any real function which satisfies the differential equation, (y’)2 + 1 = 0?

Top Courses for Mathematics

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams