Find the Domain e^(x^2)y=xe^y

Problem

e^(x^2)*y=x*e^y

Solution

1. Identify the equation as a relation between x and y where we seek the domain, which is the set of all possible values for the independent variable x

2. Analyze the functions involved in the equation. The exponential function e^u is defined for all real numbers u

3. Check for restrictions such as denominators that could be zero, square roots of negative numbers, or logarithms of non-positive numbers. None of these are present in the expression e^(x^2)*y=x*e^y

4. Determine if y can be expressed as a function of x for any x For any real value of x there exists at least one real value of y that satisfies the equation (for example, if x=0 then e^0*y=0*e^y⇒y=0.

5. Conclude that since there are no algebraic restrictions on x in the given relation, the domain consists of all real numbers.

Final Answer

Domain:(−∞,∞)

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