Question
Eliminate the arbitrary constant in the equation: X^3 -3x^2y=C
Answers
GPT-4o mini
To eliminate the arbitrary constant \( C \) from the equation \( x^3 - 3x^2y = C \), we can rearrange the equation in terms of \( y \).
Starting with the equation:
\[
x^3 - 3x^2y = C
\]
we can express \( y \) as:
\[
3x^2y = x^3 - C
\]
Now, divide both sides by \( 3x^2 \) to solve for \( y \):
\[
y = \frac{x^3 - C}{3x^2}
\]
To eliminate the constant \( C \), we take partial derivatives of \( y \) with respect to \( x \) and \( C \).
1. Differentiate \( y \) with respect to \( x \) while treating \( C \) as a constant:
\[
\frac{\partial y}{\partial x} = \frac{3x^2(1) - (x^3 - C)(2x)}{(3x^2)^2}
\]
2. Differentiate \( y \) with respect to \( C \):
\[
\frac{\partial y}{\partial C} = -\frac{1}{3x^2}
\]
Next, we can use the condition that keeps \( C \) constant and these derivatives equal. Setting the determinant (Jacobian) of the transformation formed by the variables to zero could be a way to eliminate \( C \).
In this case, we can eliminate \( C \) by considering the dependency of \( y \) on \( C \) through the derivatives. Hence, using the derivatives, we can set up a relation:
\[
3x^2 \frac{\partial y}{\partial x} + \frac{\partial y}{\partial C} = 0
\]
This will provide a differential equation independent of \( C \).
However, it is more illustrative in this case simply to recognize that the original equation itself represents a relationship between \( x \) and \( y \) that does not need C explicitly, showing its dependence.
To encapsulate everything, the resultant form shows the implicit relationship between \( x \) and \( y \):
\[
x^3 - 3x^2y = 0
\]
This can also be expressed as:
\[
y = \frac{x^3}{3x^2}
\]
or simply recognizing that:
\[
y = \frac{x}{3} \quad (x \neq 0)
\]
Thus, the arbitrary constant \( C \) has been eliminated in terms of the direct relationship between \( x \) and \( y \).
Starting with the equation:
\[
x^3 - 3x^2y = C
\]
we can express \( y \) as:
\[
3x^2y = x^3 - C
\]
Now, divide both sides by \( 3x^2 \) to solve for \( y \):
\[
y = \frac{x^3 - C}{3x^2}
\]
To eliminate the constant \( C \), we take partial derivatives of \( y \) with respect to \( x \) and \( C \).
1. Differentiate \( y \) with respect to \( x \) while treating \( C \) as a constant:
\[
\frac{\partial y}{\partial x} = \frac{3x^2(1) - (x^3 - C)(2x)}{(3x^2)^2}
\]
2. Differentiate \( y \) with respect to \( C \):
\[
\frac{\partial y}{\partial C} = -\frac{1}{3x^2}
\]
Next, we can use the condition that keeps \( C \) constant and these derivatives equal. Setting the determinant (Jacobian) of the transformation formed by the variables to zero could be a way to eliminate \( C \).
In this case, we can eliminate \( C \) by considering the dependency of \( y \) on \( C \) through the derivatives. Hence, using the derivatives, we can set up a relation:
\[
3x^2 \frac{\partial y}{\partial x} + \frac{\partial y}{\partial C} = 0
\]
This will provide a differential equation independent of \( C \).
However, it is more illustrative in this case simply to recognize that the original equation itself represents a relationship between \( x \) and \( y \) that does not need C explicitly, showing its dependence.
To encapsulate everything, the resultant form shows the implicit relationship between \( x \) and \( y \):
\[
x^3 - 3x^2y = 0
\]
This can also be expressed as:
\[
y = \frac{x^3}{3x^2}
\]
or simply recognizing that:
\[
y = \frac{x}{3} \quad (x \neq 0)
\]
Thus, the arbitrary constant \( C \) has been eliminated in terms of the direct relationship between \( x \) and \( y \).