Calculus Examples

$\frac{d y}{d t} = \frac{4 t^{2}}{3 y}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d t} = y \frac{4 t^{2}}{3 y}$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Factor $y$ out of $3 y$ .

$y \frac{d y}{d t} = y \frac{4 t^{2}}{y \cdot 3}$

Step 1.2.2

Cancel the common factor.

$y \frac{d y}{d t} = y \frac{4 t^{2}}{y \cdot 3}$

Step 1.2.3

Rewrite the expression.

$y \frac{d y}{d t} = \frac{4 t^{2}}{3}$

Step 1.3

Rewrite the equation.

$y d y = \frac{4 t^{2}}{3} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{4 t^{2}}{3} d t$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{4}{3}$ is constant with respect to $t$ , move $\frac{4}{3}$ out of the integral.

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

Step 2.3.2

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

$\frac{1}{2} y^{2} + C_{1} = \frac{4}{3} (\frac{1}{3} t^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $\frac{4}{3} (\frac{1}{3} t^{3} + C_{2})$ as $\frac{4}{3} \cdot \frac{1}{3} t^{3} + C_{2}$ .

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Multiply $\frac{4}{3}$ by $\frac{1}{3}$ .

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

Step 2.3.3.2.2

Multiply $3$ by $3$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{4}{9} t^{3} + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = \frac{4}{9} t^{3} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\frac{4}{9} t^{3} + K)$

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 $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (\frac{4}{9} t^{3} + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (\frac{4}{9} t^{3} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\frac{4}{9} t^{3} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{4}{9} t^{3} + K)$ .

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Step 3.2.2.1.1

Combine $\frac{4}{9}$ and $t^{3}$ .

$y^{2} = 2 (\frac{4 t^{3}}{9} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 \frac{4 t^{3}}{9} + 2 K$

Step 3.2.2.1.3

Multiply $2 \frac{4 t^{3}}{9}$ .

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Step 3.2.2.1.3.1

Combine $2$ and $\frac{4 t^{3}}{9}$ .

$y^{2} = \frac{2 (4 t^{3})}{9} + 2 K$

Step 3.2.2.1.3.2

Multiply $4$ by $2$ .

$y^{2} = \frac{8 t^{3}}{9} + 2 K$

Step 3.3

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

$y = \pm \sqrt{\frac{8 t^{3}}{9} + 2 K}$

Step 3.4

Simplify $\pm \sqrt{\frac{8 t^{3}}{9} + 2 K}$ .

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Step 3.4.1

Factor $2$ out of $\frac{8 t^{3}}{9} + 2 K$ .

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Step 3.4.1.1

Factor $2$ out of $\frac{8 t^{3}}{9}$ .

$y = \pm \sqrt{2 \frac{4 t^{3}}{9} + 2 K}$

Step 3.4.1.2

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 \frac{4 t^{3}}{9} + 2 K}$

Step 3.4.1.3

Factor $2$ out of $2 \frac{4 t^{3}}{9} + 2 K$ .

$y = \pm \sqrt{2 (\frac{4 t^{3}}{9} + K)}$

Step 3.4.2

To write $K$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = \pm \sqrt{2 (\frac{4 t^{3}}{9} + K \cdot \frac{9}{9})}$

Step 3.4.3

Simplify terms.

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Step 3.4.3.1

Combine $K$ and $\frac{9}{9}$ .

$y = \pm \sqrt{2 (\frac{4 t^{3}}{9} + \frac{K \cdot 9}{9})}$

Step 3.4.3.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{2 \frac{4 t^{3} + K \cdot 9}{9}}$

Step 3.4.4

Move $9$ to the left of $K$ .

$y = \pm \sqrt{2 \frac{4 t^{3} + 9 K}{9}}$

Step 3.4.5

Combine $2$ and $\frac{4 t^{3} + 9 K}{9}$ .

$y = \pm \sqrt{\frac{2 (4 t^{3} + 9 K)}{9}}$

Step 3.4.6

Rewrite $\frac{2 (4 t^{3} + 9 K)}{9}$ as ${(\frac{1}{3})}^{2} (2 (4 t^{3} + 9 K))$ .

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Step 3.4.6.1

Factor the perfect power $1^{2}$ out of $2 (4 t^{3} + 9 K)$ .

$y = \pm \sqrt{\frac{1^{2} (2 (4 t^{3} + 9 K))}{9}}$

Step 3.4.6.2

Factor the perfect power $3^{2}$ out of $9$ .

$y = \pm \sqrt{\frac{1^{2} (2 (4 t^{3} + 9 K))}{3^{2} \cdot 1}}$

Step 3.4.6.3

Rearrange the fraction $\frac{1^{2} (2 (4 t^{3} + 9 K))}{3^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{3})}^{2} (2 (4 t^{3} + 9 K))}$

Step 3.4.7

Pull terms out from under the radical.

$y = \pm \frac{1}{3} \sqrt{2 (4 t^{3} + 9 K)}$

Step 3.4.8

Combine $\frac{1}{3}$ and $\sqrt{2 (4 t^{3} + 9 K)}$ .

$y = \pm \frac{\sqrt{2 (4 t^{3} + 9 K)}}{3}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{2 (4 t^{3} + 9 K)}}{3}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{2 (4 t^{3} + 9 K)}}{3}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{2 (4 t^{3} + 9 K)}}{3}$

$y = - \frac{\sqrt{2 (4 t^{3} + 9 K)}}{3}$

$y = \frac{\sqrt{2 (4 t^{3} + 9 K)}}{3}$

$y = - \frac{\sqrt{2 (4 t^{3} + 9 K)}}{3}$

$y = \frac{\sqrt{2 (4 t^{3} + 9 K)}}{3}$

$y = - \frac{\sqrt{2 (4 t^{3} + 9 K)}}{3}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{2 (4 t^{3} + K)}}{3}$

$y = - \frac{\sqrt{2 (4 t^{3} + K)}}{3}$