Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{2} d y = \int 1 - x^{2} d x$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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

Step 2.3.6

Reorder terms.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $3 (\frac{1}{3} y^{3})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{3}$ and $y^{3}$ .

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (- \frac{1}{3} x^{3} + x + K)$ .

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Step 3.2.2.1.1

Combine $x^{3}$ and $\frac{1}{3}$ .

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.1

Move the leading negative in $- \frac{x^{3}}{3}$ into the numerator.

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

Step 3.2.2.1.3.2

Cancel the common factor.

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

Step 3.2.2.1.3.3

Rewrite the expression.

$y^{3} = - x^{3} + 3 x + 3 K$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{- x^{3} + 3 x + 3 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt[3]{- x^{3} + 3 x + K}$