Calculus Examples

$\frac{d y}{d x} = \frac{x^{2}}{y^{2}}$ , $y (0) = 2$

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$y^{2} d y = x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

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 $y^{3}$ .

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.1

Cancel the common factor.

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

Step 3.2.2.1.3.2

Rewrite the expression.

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

Step 3.3

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

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

Step 4

Simplify the constant of integration.

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

Step 5

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $2$ for $y$ in $y = \sqrt[3]{x^{3} + K}$ .

$2 = \sqrt[3]{0^{3} + K}$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $\sqrt[3]{0^{3} + K} = 2$ .

$\sqrt[3]{0^{3} + K} = 2$

Step 6.2

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{0^{3} + K}}^{3} = 2^{3}$

Step 6.3

Simplify each side of the equation.

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Step 6.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{0^{3} + K}$ as ${(0^{3} + K)}^{\frac{1}{3}}$ .

${({(0^{3} + K)}^{\frac{1}{3}})}^{3} = 2^{3}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Simplify ${({(0^{3} + K)}^{\frac{1}{3}})}^{3}$ .

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Step 6.3.2.1.1

Multiply the exponents in ${({(0^{3} + K)}^{\frac{1}{3}})}^{3}$ .

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Step 6.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(0^{3} + K)}^{\frac{1}{3} \cdot 3} = 2^{3}$

Step 6.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.3.2.1.1.2.1

Cancel the common factor.

${(0^{3} + K)}^{\frac{1}{3} \cdot 3} = 2^{3}$

Step 6.3.2.1.1.2.2

Rewrite the expression.

${(0^{3} + K)}^{1} = 2^{3}$

Step 6.3.2.1.2

Raising $0$ to any positive power yields $0$ .

${(0 + K)}^{1} = 2^{3}$

Step 6.3.2.1.3

Add $0$ and $K$ .

$K^{1} = 2^{3}$

Step 6.3.2.1.4

Simplify.

$K = 2^{3}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Raise $2$ to the power of $3$ .

$K = 8$

Step 7

Substitute $8$ for $K$ in $y = \sqrt[3]{x^{3} + K}$ and simplify.

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Step 7.1

Substitute $8$ for $K$ .

$y = \sqrt[3]{x^{3} + 8}$

Step 7.2

Rewrite $8$ as $2^{3}$ .

$y = \sqrt[3]{x^{3} + 2^{3}}$

Step 7.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 2$ .

$y = \sqrt[3]{(x + 2) (x^{2} - x \cdot 2 + 2^{2})}$

Step 7.4

Simplify.

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Step 7.4.1

Multiply $2$ by $- 1$ .

$y = \sqrt[3]{(x + 2) (x^{2} - 2 x + 2^{2})}$

Step 7.4.2

Raise $2$ to the power of $2$ .

$y = \sqrt[3]{(x + 2) (x^{2} - 2 x + 4)}$