Calculus Examples

$7 y \frac{d y}{d x} = 6 x^{2}$

Step 1

Rewrite the equation.

$7 y d y = 6 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 7 y d y = \int 6 x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

$7 \int y d y = \int 6 x^{2} d x$

Step 2.2.2

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

$7 (\frac{1}{2} y^{2} + C_{1}) = \int 6 x^{2} d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

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

$7 (\frac{1}{2}) y^{2} + C_{1} = \int 6 x^{2} d x$

Step 2.2.3.2

Combine $7$ and $\frac{1}{2}$ .

$\frac{7}{2} y^{2} + C_{1} = \int 6 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{7}{2} y^{2} + C_{1} = 6 \int x^{2} d x$

Step 2.3.2

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

$\frac{7}{2} y^{2} + C_{1} = 6 (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $6 (\frac{1}{3} x^{3} + C_{2})$ as $6 (\frac{1}{3}) x^{3} + C_{2}$ .

$\frac{7}{2} y^{2} + C_{1} = 6 (\frac{1}{3}) x^{3} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $6$ and $\frac{1}{3}$ .

$\frac{7}{2} y^{2} + C_{1} = \frac{6}{3} x^{3} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $6$ and $3$ .

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Step 2.3.3.2.2.1

Factor $3$ out of $6$ .

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

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $3$ out of $3$ .

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

Step 2.3.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.3.2.2.2.4

Divide $2$ by $1$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $\frac{2}{7}$ .

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

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 $\frac{2}{7} (\frac{7}{2} y^{2})$ .

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Step 3.2.1.1.1

Combine $\frac{7}{2}$ and $y^{2}$ .

$\frac{2}{7} \cdot \frac{7 y^{2}}{2} = \frac{2}{7} (2 x^{3} + K)$

Step 3.2.1.1.2

Combine.

$\frac{2 (7 y^{2})}{7 \cdot 2} = \frac{2}{7} (2 x^{3} + K)$

Step 3.2.1.1.3

Cancel the common factor of $2$ .

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Step 3.2.1.1.3.1

Cancel the common factor.

$\frac{2 (7 y^{2})}{7 \cdot 2} = \frac{2}{7} (2 x^{3} + K)$

Step 3.2.1.1.3.2

Rewrite the expression.

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

Step 3.2.1.1.4

Cancel the common factor of $7$ .

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Step 3.2.1.1.4.1

Cancel the common factor.

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

Step 3.2.1.1.4.2

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

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{2}{7} (2 x^{3} + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

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

Step 3.2.2.1.2

Multiply $\frac{2}{7} (2 x^{3})$ .

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Step 3.2.2.1.2.1

Combine $2$ and $\frac{2}{7}$ .

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

Step 3.2.2.1.2.2

Multiply $2$ by $2$ .

$y^{2} = \frac{4}{7} x^{3} + \frac{2}{7} K$

Step 3.2.2.1.2.3

Combine $\frac{4}{7}$ and $x^{3}$ .

$y^{2} = \frac{4 x^{3}}{7} + \frac{2}{7} K$

Step 3.2.2.1.3

Combine $\frac{2}{7}$ and $K$ .

$y^{2} = \frac{4 x^{3}}{7} + \frac{2 K}{7}$

Step 3.3

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

$y = \pm \sqrt{\frac{4 x^{3}}{7} + \frac{2 K}{7}}$

Step 3.4

Simplify $\pm \sqrt{\frac{4 x^{3}}{7} + \frac{2 K}{7}}$ .

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Step 3.4.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{4 x^{3} + 2 K}{7}}$

Step 3.4.2

Factor $2$ out of $4 x^{3} + 2 K$ .

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Step 3.4.2.1

Factor $2$ out of $4 x^{3}$ .

$y = \pm \sqrt{\frac{2 (2 x^{3}) + 2 K}{7}}$

Step 3.4.2.2

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{\frac{2 (2 x^{3}) + 2 K}{7}}$

Step 3.4.2.3

Factor $2$ out of $2 (2 x^{3}) + 2 K$ .

$y = \pm \sqrt{\frac{2 (2 x^{3} + K)}{7}}$

Step 3.4.3

Rewrite $\sqrt{\frac{2 (2 x^{3} + K)}{7}}$ as $\frac{\sqrt{2 (2 x^{3} + K)}}{\sqrt{7}}$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)}}{\sqrt{7}}$

Step 3.4.4

Multiply $\frac{\sqrt{2 (2 x^{3} + K)}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 3.4.5

Combine and simplify the denominator.

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Step 3.4.5.1

Multiply $\frac{\sqrt{2 (2 x^{3} + K)}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.4.5.2

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 3.4.5.3

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 3.4.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 3.4.5.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 3.4.5.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 3.4.5.6.2

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

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 3.4.5.6.3

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

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.4.5.6.4

Cancel the common factor of $2$ .

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Step 3.4.5.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.4.5.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{7^{1}}$

Step 3.4.5.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{2 (2 x^{3} + K)} \sqrt{7}}{7}$

Step 3.4.6

Simplify the numerator.

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Step 3.4.6.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{2 (2 x^{3} + K) \cdot 7}}{7}$

Step 3.4.6.2

Multiply $7$ by $2$ .

$y = \pm \frac{\sqrt{14 (2 x^{3} + K)}}{7}$

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.

$y = \frac{\sqrt{14 (2 x^{3} + K)}}{7}$

Step 3.5.2

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

$y = - \frac{\sqrt{14 (2 x^{3} + K)}}{7}$

Step 3.5.3

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

$y = \frac{\sqrt{14 (2 x^{3} + K)}}{7}$

$y = - \frac{\sqrt{14 (2 x^{3} + K)}}{7}$

$y = \frac{\sqrt{14 (2 x^{3} + K)}}{7}$

$y = - \frac{\sqrt{14 (2 x^{3} + K)}}{7}$

$y = \frac{\sqrt{14 (2 x^{3} + K)}}{7}$

$y = - \frac{\sqrt{14 (2 x^{3} + K)}}{7}$