Calculus Examples

$x^{2} - (y^{3} + 2) \frac{d y}{d x} = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

$x^{2} + (- y^{3} - 1 \cdot 2) \frac{d y}{d x} = 0$

Step 1.1.1.2

Multiply $- 1$ by $2$ .

$x^{2} + (- y^{3} - 2) \frac{d y}{d x} = 0$

Step 1.1.1.3

Apply the distributive property.

$x^{2} - y^{3} \frac{d y}{d x} - 2 \frac{d y}{d x} = 0$

Step 1.1.2

Subtract $x^{2}$ from both sides of the equation.

$- y^{3} \frac{d y}{d x} - 2 \frac{d y}{d x} = - x^{2}$

Step 1.1.3

Factor $\frac{d y}{d x}$ out of $- y^{3} \frac{d y}{d x} - 2 \frac{d y}{d x}$ .

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Step 1.1.3.1

Factor $\frac{d y}{d x}$ out of $- y^{3} \frac{d y}{d x}$ .

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

Step 1.1.3.2

Factor $\frac{d y}{d x}$ out of $- 2 \frac{d y}{d x}$ .

$\frac{d y}{d x} (- y^{3}) + \frac{d y}{d x} \cdot - 2 = - x^{2}$

Step 1.1.3.3

Factor $\frac{d y}{d x}$ out of $\frac{d y}{d x} (- y^{3}) + \frac{d y}{d x} \cdot - 2$ .

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

Step 1.1.4

Divide each term in $\frac{d y}{d x} (- y^{3} - 2) = - x^{2}$ by $- y^{3} - 2$ and simplify.

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Step 1.1.4.1

Divide each term in $\frac{d y}{d x} (- y^{3} - 2) = - x^{2}$ by $- y^{3} - 2$ .

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

Step 1.1.4.2

Simplify the left side.

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Step 1.1.4.2.1

Cancel the common factor of $- y^{3} - 2$ .

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Step 1.1.4.2.1.1

Cancel the common factor.

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

Step 1.1.4.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{- x^{2}}{- y^{3} - 2}$

Step 1.1.4.3

Simplify the right side.

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Step 1.1.4.3.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{x^{2}}{- y^{3} - 2}$

Step 1.1.4.3.2

Factor $- 1$ out of $- y^{3}$ .

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

Step 1.1.4.3.3

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{d y}{d x} = - \frac{x^{2}}{- (y^{3}) - 1 (2)}$

Step 1.1.4.3.4

Factor $- 1$ out of $- (y^{3}) - 1 (2)$ .

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

Step 1.1.4.3.5

Simplify the expression.

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Step 1.1.4.3.5.1

Rewrite $- (y^{3} + 2)$ as $- 1 (y^{3} + 2)$ .

$\frac{d y}{d x} = - \frac{x^{2}}{- 1 (y^{3} + 2)}$

Step 1.1.4.3.5.2

Move the negative in front of the fraction.

$\frac{d y}{d x} = - - \frac{x^{2}}{y^{3} + 2}$

Step 1.1.4.3.5.3

Multiply $- 1$ by $- 1$ .

$\frac{d y}{d x} = 1 \frac{x^{2}}{y^{3} + 2}$

Step 1.1.4.3.5.4

Multiply $\frac{x^{2}}{y^{3} + 2}$ by $1$ .

$\frac{d y}{d x} = \frac{x^{2}}{y^{3} + 2}$

Step 1.2

Multiply both sides by $y^{3} + 2$ .

$(y^{3} + 2) \frac{d y}{d x} = (y^{3} + 2) \frac{x^{2}}{y^{3} + 2}$

Step 1.3

Cancel the common factor of $y^{3} + 2$ .

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Step 1.3.1

Cancel the common factor.

$(y^{3} + 2) \frac{d y}{d x} = (y^{3} + 2) \frac{x^{2}}{y^{3} + 2}$

Step 1.3.2

Rewrite the expression.

$(y^{3} + 2) \frac{d y}{d x} = x^{2}$

Step 1.4

Rewrite the equation.

$(y^{3} + 2) d y = x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{3} + 2 d y = \int x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int y^{3} d y + \int 2 d y = \int x^{2} d x$

Step 2.2.2

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

$\frac{1}{4} y^{4} + C_{1} + \int 2 d y = \int x^{2} d x$

Step 2.2.3

Apply the constant rule.

$\frac{1}{4} y^{4} + C_{1} + 2 y + C_{2} = \int x^{2} d x$

Step 2.2.4

Simplify.

$\frac{1}{4} y^{4} + 2 y + C_{3} = \int x^{2} d x$

Step 2.3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{4} y^{4} + 2 y + C_{3} = \frac{1}{3} x^{3} + C_{4}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{4} y^{4} + 2 y = \frac{1}{3} x^{3} + K$