Calculus Examples

$2 \frac{d y}{d x} = \frac{7 x^{3}}{y}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y (2 \frac{d y}{d x}) = y \frac{7 x^{3}}{y}$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Cancel the common factor.

$y (2 \frac{d y}{d x}) = y \frac{7 x^{3}}{y}$

Step 1.2.2

Rewrite the expression.

$y (2 \frac{d y}{d x}) = 7 x^{3}$

Step 1.3

Remove unnecessary parentheses.

$y \cdot 2 \frac{d y}{d x} = 7 x^{3}$

Step 1.4

Rewrite the equation.

$y \cdot 2 d y = 7 x^{3} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y \cdot 2 d y = \int 7 x^{3} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Move $2$ to the left of $y$ .

$\int 2 y d y = \int 7 x^{3} d x$

Step 2.2.2

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$2 \int y d y = \int 7 x^{3} d x$

Step 2.2.3

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

$2 (\frac{1}{2} y^{2} + C_{1}) = \int 7 x^{3} d x$

Step 2.2.4

Simplify the answer.

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Step 2.2.4.1

Rewrite $2 (\frac{1}{2} y^{2} + C_{1})$ as $2 (\frac{1}{2}) y^{2} + C_{1}$ .

$2 (\frac{1}{2}) y^{2} + C_{1} = \int 7 x^{3} d x$

Step 2.2.4.2

Simplify.

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Step 2.2.4.2.1

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

$\frac{2}{2} y^{2} + C_{1} = \int 7 x^{3} d x$

Step 2.2.4.2.2

Cancel the common factor of $2$ .

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Step 2.2.4.2.2.1

Cancel the common factor.

$\frac{2}{2} y^{2} + C_{1} = \int 7 x^{3} d x$

Step 2.2.4.2.2.2

Rewrite the expression.

$1 y^{2} + C_{1} = \int 7 x^{3} d x$

Step 2.2.4.2.3

Multiply $y^{2}$ by $1$ .

$y^{2} + C_{1} = \int 7 x^{3} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$y^{2} + C_{1} = 7 \int x^{3} d x$

Step 2.3.2

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

$y^{2} + C_{1} = 7 (\frac{1}{4} x^{4} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $7 (\frac{1}{4} x^{4} + C_{2})$ as $7 (\frac{1}{4}) x^{4} + C_{2}$ .

$y^{2} + C_{1} = 7 (\frac{1}{4}) x^{4} + C_{2}$

Step 2.3.3.2

Combine $7$ and $\frac{1}{4}$ .

$y^{2} + C_{1} = \frac{7}{4} x^{4} + C_{2}$

Step 2.4

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

$y^{2} = \frac{7}{4} x^{4} + K$

Step 3

Solve for $y$ .

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Step 3.1

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

$y = \pm \sqrt{\frac{7}{4} x^{4} + K}$

Step 3.2

Simplify $\pm \sqrt{\frac{7}{4} x^{4} + K}$ .

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Step 3.2.1

Combine $\frac{7}{4}$ and $x^{4}$ .

$y = \pm \sqrt{\frac{7 x^{4}}{4} + K}$

Step 3.2.2

To write $K$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \pm \sqrt{\frac{7 x^{4}}{4} + K \cdot \frac{4}{4}}$

Step 3.2.3

Simplify terms.

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Step 3.2.3.1

Combine $K$ and $\frac{4}{4}$ .

$y = \pm \sqrt{\frac{7 x^{4}}{4} + \frac{K \cdot 4}{4}}$

Step 3.2.3.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{7 x^{4} + K \cdot 4}{4}}$

Step 3.2.4

Move $4$ to the left of $K$ .

$y = \pm \sqrt{\frac{7 x^{4} + 4 K}{4}}$

Step 3.2.5

Rewrite $\frac{7 x^{4} + 4 K}{4}$ as ${(\frac{1}{2})}^{2} (7 x^{4} + 4 K)$ .

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Step 3.2.5.1

Factor the perfect power $1^{2}$ out of $7 x^{4} + 4 K$ .

$y = \pm \sqrt{\frac{1^{2} (7 x^{4} + 4 K)}{4}}$

Step 3.2.5.2

Factor the perfect power $2^{2}$ out of $4$ .

$y = \pm \sqrt{\frac{1^{2} (7 x^{4} + 4 K)}{2^{2} \cdot 1}}$

Step 3.2.5.3

Rearrange the fraction $\frac{1^{2} (7 x^{4} + 4 K)}{2^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{2})}^{2} (7 x^{4} + 4 K)}$

Step 3.2.6

Pull terms out from under the radical.

$y = \pm \frac{1}{2} \sqrt{7 x^{4} + 4 K}$

Step 3.2.7

Combine $\frac{1}{2}$ and $\sqrt{7 x^{4} + 4 K}$ .

$y = \pm \frac{\sqrt{7 x^{4} + 4 K}}{2}$

Step 3.3

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

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Step 3.3.1

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

$y = \frac{\sqrt{7 x^{4} + 4 K}}{2}$

Step 3.3.2

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

$y = - \frac{\sqrt{7 x^{4} + 4 K}}{2}$

Step 3.3.3

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

$y = \frac{\sqrt{7 x^{4} + 4 K}}{2}$

$y = - \frac{\sqrt{7 x^{4} + 4 K}}{2}$

$y = \frac{\sqrt{7 x^{4} + 4 K}}{2}$

$y = - \frac{\sqrt{7 x^{4} + 4 K}}{2}$

$y = \frac{\sqrt{7 x^{4} + 4 K}}{2}$

$y = - \frac{\sqrt{7 x^{4} + 4 K}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{7 x^{4} + K}}{2}$

$y = - \frac{\sqrt{7 x^{4} + K}}{2}$