Solve y^3 - 3xy^2 = x^33x^2y

Question

Solve the equation

y = 0 y = \frac{3 x + x 9 + 12 x ^{3}}{2} y = \frac{3 x - x 9 + 12 x ^{3}}{2}

Show Solution

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

Show Solution

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{5 y x ^{4} + y ^{2}}{y ^{2} - 2 x y - x ^{5}}

Calculate

y^{3} - 3 x y^{2} = x^{3} 3 x^{2} y

Simplify the expression

y^{3} - 3 x y^{2} = 3 x^{5} y

Take the derivative of both sides

\frac{d}{d x} (y^{3} - 3 x y^{2}) = \frac{d}{d x} (3 x^{5} y)

Calculate the derivative

More Steps

3 y^{2} \frac{d y}{d x} - 3 y^{2} - 6 x y \frac{d y}{d x} = \frac{d}{d x} (3 x^{5} y)

Calculate the derivative

More Steps

3 y^{2} \frac{d y}{d x} - 3 y^{2} - 6 x y \frac{d y}{d x} = 15 x^{4} y + 3 x^{5} \frac{d y}{d x}

Collect like terms by calculating the sum or difference of their coefficients

(3 y^{2} - 6 x y) \frac{d y}{d x} - 3 y^{2} = 15 x^{4} y + 3 x^{5} \frac{d y}{d x}

Move the expression to the left side

(3 y^{2} - 6 x y) \frac{d y}{d x} - 3 y^{2} - 3 x^{5} \frac{d y}{d x} = 15 x^{4} y

Move the expression to the right side

(3 y^{2} - 6 x y) \frac{d y}{d x} - 3 x^{5} \frac{d y}{d x} = 15 x^{4} y + 3 y^{2}

Collect like terms by calculating the sum or difference of their coefficients

(3 y^{2} - 6 x y - 3 x^{5}) \frac{d y}{d x} = 15 x^{4} y + 3 y^{2}

Divide both sides

\frac{( 3 y ^{2} - 6 x y - 3 x ^{5} ) \frac{d y}{d x}}{3 y ^{2} - 6 x y - 3 x ^{5}} = \frac{15 x ^{4} y + 3 y ^{2}}{3 y ^{2} - 6 x y - 3 x ^{5}}

Divide the numbers

\frac{d y}{d x} = \frac{15 x ^{4} y + 3 y ^{2}}{3 y ^{2} - 6 x y - 3 x ^{5}}

Solution

More Steps

\frac{d y}{d x} = \frac{5 y x ^{4} + y ^{2}}{y ^{2} - 2 x y - x ^{5}}

Show Solution