Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Divide each term in $(y^{2} + x^{2} y^{2}) \frac{d y}{d x} = 1$ by $y^{2} + x^{2} y^{2}$ and simplify.

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Step 1.1.1

Divide each term in $(y^{2} + x^{2} y^{2}) \frac{d y}{d x} = 1$ by $y^{2} + x^{2} y^{2}$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

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

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Factor $y^{2}$ out of $y^{2} + x^{2} y^{2}$ .

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Step 1.1.3.1.1

Multiply by $1$ .

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

Step 1.1.3.1.2

Factor $y^{2}$ out of $x^{2} y^{2}$ .

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

Step 1.1.3.1.3

Factor $y^{2}$ out of $y^{2} \cdot 1 + y^{2} x^{2}$ .

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

Step 1.2

Regroup factors.

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

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

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Step 1.4.2.1

Cancel the common factor.

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

Step 1.4.2.2

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.3.2

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C_{2}$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

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 $y^{3}$ .

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

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

Step 3.3

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

$y = \sqrt[3]{3 \arctan (x) + 3 K}$

Step 3.4

Factor $3$ out of $3 \arctan (x) + 3 K$ .

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Step 3.4.1

Factor $3$ out of $3 \arctan (x)$ .

$y = \sqrt[3]{3 (\arctan (x)) + 3 K}$

Step 3.4.2

Factor $3$ out of $3 (\arctan (x)) + 3 K$ .

$y = \sqrt[3]{3 (\arctan (x) + K)}$