Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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.

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

Step 2.2.2

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

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

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

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

Step 2.2.3.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Combine $2$ and $\frac{1}{3}$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Combine.

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

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.

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

Step 3.2.1.1.3.2

Rewrite the expression.

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

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.

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

Step 3.2.1.1.4.2

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

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Simplify terms.

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Step 3.2.2.1.1.1

Combine $\frac{2}{3}$ and $x^{3}$ .

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

Step 3.2.2.1.1.2

Apply the distributive property.

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

Step 3.2.2.1.1.3

Combine.

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

Step 3.2.2.1.1.4

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

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

Step 3.2.2.1.2

Simplify each term.

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Step 3.2.2.1.2.1

Multiply $2$ by $2$ .

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

Step 3.2.2.1.2.2

Multiply $7$ by $3$ .

$y^{2} = \frac{2^{2} x^{3}}{21} + \frac{2 K}{7}$

Step 3.2.2.1.2.3

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

$y^{2} = \frac{4 x^{3}}{21} + \frac{2 K}{7}$

Step 3.3

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

$y = \pm \sqrt{\frac{4 x^{3}}{21} + \frac{2 K}{7}}$

Step 3.4

Simplify $\pm \sqrt{\frac{4 x^{3}}{21} + \frac{2 K}{7}}$ .

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Step 3.4.1

To write $\frac{2 K}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{4 x^{3}}{21} + \frac{2 K}{7} \cdot \frac{3}{3}}$

Step 3.4.2

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.1

Multiply $\frac{2 K}{7}$ by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{4 x^{3}}{21} + \frac{2 K \cdot 3}{7 \cdot 3}}$

Step 3.4.2.2

Multiply $7$ by $3$ .

$y = \pm \sqrt{\frac{4 x^{3}}{21} + \frac{2 K \cdot 3}{21}}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{4 x^{3} + 2 K \cdot 3}{21}}$

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

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

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Step 3.4.4.1.1

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

$y = \pm \sqrt{\frac{2 (2 x^{3}) + 2 K \cdot 3}{21}}$

Step 3.4.4.1.2

Factor $2$ out of $2 K \cdot 3$ .

$y = \pm \sqrt{\frac{2 (2 x^{3}) + 2 (K \cdot 3)}{21}}$

Step 3.4.4.1.3

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

$y = \pm \sqrt{\frac{2 (2 x^{3} + K \cdot 3)}{21}}$

Step 3.4.4.2

Move $3$ to the left of $K$ .

$y = \pm \sqrt{\frac{2 (2 x^{3} + 3 K)}{21}}$

Step 3.4.5

Rewrite $\sqrt{\frac{2 (2 x^{3} + 3 K)}{21}}$ as $\frac{\sqrt{2 (2 x^{3} + 3 K)}}{\sqrt{21}}$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)}}{\sqrt{21}}$

Step 3.4.6

Multiply $\frac{\sqrt{2 (2 x^{3} + 3 K)}}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)}}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}}$

Step 3.4.7

Combine and simplify the denominator.

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Step 3.4.7.1

Multiply $\frac{\sqrt{2 (2 x^{3} + 3 K)}}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 3.4.7.2

Raise $\sqrt{21}$ to the power of $1$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{{\sqrt{21}}^{1} \sqrt{21}}$

Step 3.4.7.3

Raise $\sqrt{21}$ to the power of $1$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{{\sqrt{21}}^{1} {\sqrt{21}}^{1}}$

Step 3.4.7.4

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

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{{\sqrt{21}}^{1 + 1}}$

Step 3.4.7.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{{\sqrt{21}}^{2}}$

Step 3.4.7.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 3.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}$

Step 3.4.7.6.2

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

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{21^{\frac{1}{2} \cdot 2}}$

Step 3.4.7.6.3

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

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{21^{\frac{2}{2}}}$

Step 3.4.7.6.4

Cancel the common factor of $2$ .

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Step 3.4.7.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{21^{\frac{2}{2}}}$

Step 3.4.7.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{21^{1}}$

Step 3.4.7.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K)} \sqrt{21}}{21}$

Step 3.4.8

Simplify the numerator.

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Step 3.4.8.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{2 (2 x^{3} + 3 K) \cdot 21}}{21}$

Step 3.4.8.2

Multiply $21$ by $2$ .

$y = \pm \frac{\sqrt{42 (2 x^{3} + 3 K)}}{21}$

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.

$y = \frac{\sqrt{42 (2 x^{3} + 3 K)}}{21}$

Step 3.5.2

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

$y = - \frac{\sqrt{42 (2 x^{3} + 3 K)}}{21}$

Step 3.5.3

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

$y = \frac{\sqrt{42 (2 x^{3} + 3 K)}}{21}$

$y = - \frac{\sqrt{42 (2 x^{3} + 3 K)}}{21}$

$y = \frac{\sqrt{42 (2 x^{3} + 3 K)}}{21}$

$y = - \frac{\sqrt{42 (2 x^{3} + 3 K)}}{21}$

$y = \frac{\sqrt{42 (2 x^{3} + 3 K)}}{21}$

$y = - \frac{\sqrt{42 (2 x^{3} + 3 K)}}{21}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{42 (2 x^{3} + K)}}{21}$

$y = - \frac{\sqrt{42 (2 x^{3} + K)}}{21}$