Find the Domain

Problem

(2*x)/(y3)*d(x)+(y2+3(x2))/(y4)*d(y)=0

Solution

1. Identify the components of the differential equation. The expression is given in the form M(x,y)*d(x)+N(x,y)*d(y)=0 where M(x,y)=(2*x)/(y3) and N(x,y)=(y2+3(x2))/(y4)

2. Analyze the constraints for the function M(x,y) The term (2*x)/(y3) is defined for all real values of x and y provided that the denominator is not zero.

3. Set the condition for the denominator of M(x,y)

y3≠0

⇒y≠0

4. Analyze the constraints for the function N(x,y) The term (y2+3(x2))/(y4) involves an exponential function 3(x2) which is defined for all real x The fraction is defined provided that the denominator is not zero.

5. Set the condition for the denominator of N(x,y)

y4≠0

⇒y≠0

6. Combine the restrictions. The only restriction for the entire expression to be defined is that y cannot be zero. There are no restrictions on x

Final Answer

Domain={(x,y)∈ℝ2:y≠0}

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