Calculus Examples

$3 y^{2} \frac{d y}{d x} + x^{- 1} y^{3} = x^{4}$

Step 1

Let $v = y^{3}$ . Substitute $v$ for all occurrences of $y^{3}$ .

$3 y^{2} \frac{d y}{d x} + x^{- 1} v = x^{4}$

Step 2

Find $\frac{d v}{d x}$ by differentiating $y^{3}$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $y$ .

$\frac{d v}{d x} = \frac{d}{d u} [u^{3}] \frac{d}{d x} [y]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$\frac{d v}{d x} = 3 u^{2} \frac{d}{d x} [y]$

Step 2.1.3

Replace all occurrences of $u$ with $y$ .

$\frac{d v}{d x} = 3 y^{2} \frac{d}{d x} [y]$

Step 2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

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

Step 3

Substitute the derivative back in to the differential equation.

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Step 3.1

Cancel the common factor of $3 y^{2}$ .

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Step 3.1.1

Cancel the common factor.

$3 y^{2} \frac{\frac{d v}{d x}}{3 y^{2}} + x^{- 1} v = x^{4}$

Step 3.1.2

Rewrite the expression.

$\frac{d v}{d x} + x^{- 1} v = x^{4}$

Step 3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d v}{d x} + \frac{1}{x} v = x^{4}$

Step 3.3

Combine $\frac{1}{x}$ and $v$ .

$\frac{d v}{d x} + \frac{v}{x} = x^{4}$

Step 4

Rewrite the differential equation as $\frac{d v}{d x} + M (x) v = Q (x)$ .

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Step 4.1

Factor $v$ out of $\frac{v}{x}$ .

$\frac{d v}{d x} + v \frac{1}{x} = x^{4}$

Step 4.2

Reorder $v$ and $\frac{1}{x}$ .

$\frac{d v}{d x} + \frac{1}{x} v = x^{4}$

Step 5

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{x}$ .

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Step 5.1

Set up the integration.

$e^{\int \frac{1}{x} d x}$

Step 5.2

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$e^{\ln (| x |) + C}$

Step 5.3

Remove the constant of integration.

$e^{\ln (x)}$

Step 5.4

Exponentiation and log are inverse functions.

$x$

Step 6

Multiply each term by the integrating factor $x$ .

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Step 6.1

Multiply each term by $x$ .

$x \frac{d v}{d x} + x (\frac{1}{x} v) = x \cdot x^{4}$

Step 6.2

Simplify each term.

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Step 6.2.1

Combine $\frac{1}{x}$ and $v$ .

$x \frac{d v}{d x} + x \frac{v}{x} = x \cdot x^{4}$

Step 6.2.2

Cancel the common factor of $x$ .

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Step 6.2.2.1

Cancel the common factor.

$x \frac{d v}{d x} + x \frac{v}{x} = x \cdot x^{4}$

Step 6.2.2.2

Rewrite the expression.

$x \frac{d v}{d x} + v = x \cdot x^{4}$

Step 6.3

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 6.3.1

Multiply $x$ by $x^{4}$ .

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Step 6.3.1.1

Raise $x$ to the power of $1$ .

$x \frac{d v}{d x} + v = x^{1} x^{4}$

Step 6.3.1.2

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

$x \frac{d v}{d x} + v = x^{1 + 4}$

Step 6.3.2

Add $1$ and $4$ .

$x \frac{d v}{d x} + v = x^{5}$

Step 7

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [x v] = x^{5}$

Step 8

Set up an integral on each side.

$\int \frac{d}{d x} [x v] d x = \int x^{5} d x$

Step 9

Integrate the left side.

$x v = \int x^{5} d x$

Step 10

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$x v = \frac{1}{6} x^{6} + C$

Step 11

Divide each term in $x v = \frac{1}{6} x^{6} + C$ by $x$ and simplify.

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Step 11.1

Divide each term in $x v = \frac{1}{6} x^{6} + C$ by $x$ .

$\frac{x v}{x} = \frac{\frac{1}{6} x^{6}}{x} + \frac{C}{x}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $x$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{x v}{x} = \frac{\frac{1}{6} x^{6}}{x} + \frac{C}{x}$

Step 11.2.1.2

Divide $v$ by $1$ .

$v = \frac{\frac{1}{6} x^{6}}{x} + \frac{C}{x}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Simplify each term.

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Step 11.3.1.1

Cancel the common factor of $x^{6}$ and $x$ .

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Step 11.3.1.1.1

Factor $x$ out of $\frac{1}{6} x^{6}$ .

$v = \frac{x (\frac{1}{6} x^{5})}{x} + \frac{C}{x}$

Step 11.3.1.1.2

Cancel the common factors.

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Step 11.3.1.1.2.1

Raise $x$ to the power of $1$ .

$v = \frac{x (\frac{1}{6} x^{5})}{x^{1}} + \frac{C}{x}$

Step 11.3.1.1.2.2

Factor $x$ out of $x^{1}$ .

$v = \frac{x (\frac{1}{6} x^{5})}{x \cdot 1} + \frac{C}{x}$

Step 11.3.1.1.2.3

Cancel the common factor.

$v = \frac{x (\frac{1}{6} x^{5})}{x \cdot 1} + \frac{C}{x}$

Step 11.3.1.1.2.4

Rewrite the expression.

$v = \frac{\frac{1}{6} x^{5}}{1} + \frac{C}{x}$

Step 11.3.1.1.2.5

Divide $\frac{1}{6} x^{5}$ by $1$ .

$v = \frac{1}{6} x^{5} + \frac{C}{x}$

Step 11.3.1.2

Combine $\frac{1}{6}$ and $x^{5}$ .

$v = \frac{x^{5}}{6} + \frac{C}{x}$

Step 12

Replace all occurrences of $v$ with $y^{3}$ .

$y^{3} = \frac{x^{5}}{6} + \frac{C}{x}$

Step 13

Solve for $y$ .

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Step 13.1

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

$y = \sqrt[3]{\frac{x^{5}}{6} + \frac{C}{x}}$

Step 13.2

Simplify $\sqrt[3]{\frac{x^{5}}{6} + \frac{C}{x}}$ .

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Step 13.2.1

To write $\frac{x^{5}}{6}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y = \sqrt[3]{\frac{x^{5}}{6} \cdot \frac{x}{x} + \frac{C}{x}}$

Step 13.2.2

To write $\frac{C}{x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = \sqrt[3]{\frac{x^{5}}{6} \cdot \frac{x}{x} + \frac{C}{x} \cdot \frac{6}{6}}$

Step 13.2.3

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.2.3.1

Multiply $\frac{x^{5}}{6}$ by $\frac{x}{x}$ .

$y = \sqrt[3]{\frac{x^{5} x}{6 x} + \frac{C}{x} \cdot \frac{6}{6}}$

Step 13.2.3.2

Multiply $\frac{C}{x}$ by $\frac{6}{6}$ .

$y = \sqrt[3]{\frac{x^{5} x}{6 x} + \frac{C \cdot 6}{x \cdot 6}}$

Step 13.2.3.3

Reorder the factors of $x \cdot 6$ .

$y = \sqrt[3]{\frac{x^{5} x}{6 x} + \frac{C \cdot 6}{6 x}}$

Step 13.2.4

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{x^{5} x + C \cdot 6}{6 x}}$

Step 13.2.5

Simplify the numerator.

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Step 13.2.5.1

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 13.2.5.1.1

Multiply $x^{5}$ by $x$ .

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Step 13.2.5.1.1.1

Raise $x$ to the power of $1$ .

$y = \sqrt[3]{\frac{x^{5} x^{1} + C \cdot 6}{6 x}}$

Step 13.2.5.1.1.2

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

$y = \sqrt[3]{\frac{x^{5 + 1} + C \cdot 6}{6 x}}$

Step 13.2.5.1.2

Add $5$ and $1$ .

$y = \sqrt[3]{\frac{x^{6} + C \cdot 6}{6 x}}$

Step 13.2.5.2

Move $6$ to the left of $C$ .

$y = \sqrt[3]{\frac{x^{6} + 6 C}{6 x}}$

Step 13.2.6

Rewrite $\sqrt[3]{\frac{x^{6} + 6 C}{6 x}}$ as $\frac{\sqrt[3]{x^{6} + 6 C}}{\sqrt[3]{6 x}}$ .

$y = \frac{\sqrt[3]{x^{6} + 6 C}}{\sqrt[3]{6 x}}$

Step 13.2.7

Multiply $\frac{\sqrt[3]{x^{6} + 6 C}}{\sqrt[3]{6 x}}$ by $\frac{{\sqrt[3]{6 x}}^{2}}{{\sqrt[3]{6 x}}^{2}}$ .

$y = \frac{\sqrt[3]{x^{6} + 6 C}}{\sqrt[3]{6 x}} \cdot \frac{{\sqrt[3]{6 x}}^{2}}{{\sqrt[3]{6 x}}^{2}}$

Step 13.2.8

Combine and simplify the denominator.

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Step 13.2.8.1

Multiply $\frac{\sqrt[3]{x^{6} + 6 C}}{\sqrt[3]{6 x}}$ by $\frac{{\sqrt[3]{6 x}}^{2}}{{\sqrt[3]{6 x}}^{2}}$ .

$y = \frac{\sqrt[3]{x^{6} + 6 C} {\sqrt[3]{6 x}}^{2}}{\sqrt[3]{6 x} {\sqrt[3]{6 x}}^{2}}$

Step 13.2.8.2

Raise $\sqrt[3]{6 x}$ to the power of $1$ .

$y = \frac{\sqrt[3]{x^{6} + 6 C} {\sqrt[3]{6 x}}^{2}}{{\sqrt[3]{6 x}}^{1} {\sqrt[3]{6 x}}^{2}}$

Step 13.2.8.3

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

$y = \frac{\sqrt[3]{x^{6} + 6 C} {\sqrt[3]{6 x}}^{2}}{{\sqrt[3]{6 x}}^{1 + 2}}$

Step 13.2.8.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{x^{6} + 6 C} {\sqrt[3]{6 x}}^{2}}{{\sqrt[3]{6 x}}^{3}}$

Step 13.2.8.5

Rewrite ${\sqrt[3]{6 x}}^{3}$ as $6 x$ .

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Step 13.2.8.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6 x}$ as ${(6 x)}^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{x^{6} + 6 C} {\sqrt[3]{6 x}}^{2}}{{({(6 x)}^{\frac{1}{3}})}^{3}}$

Step 13.2.8.5.2

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

$y = \frac{\sqrt[3]{x^{6} + 6 C} {\sqrt[3]{6 x}}^{2}}{{(6 x)}^{\frac{1}{3} \cdot 3}}$

Step 13.2.8.5.3

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

$y = \frac{\sqrt[3]{x^{6} + 6 C} {\sqrt[3]{6 x}}^{2}}{{(6 x)}^{\frac{3}{3}}}$

Step 13.2.8.5.4

Cancel the common factor of $3$ .

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Step 13.2.8.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{x^{6} + 6 C} {\sqrt[3]{6 x}}^{2}}{{(6 x)}^{\frac{3}{3}}}$

Step 13.2.8.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x^{6} + 6 C} {\sqrt[3]{6 x}}^{2}}{{(6 x)}^{1}}$

Step 13.2.8.5.5

Simplify.

$y = \frac{\sqrt[3]{x^{6} + 6 C} {\sqrt[3]{6 x}}^{2}}{6 x}$

Step 13.2.9

Simplify the numerator.

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Step 13.2.9.1

Rewrite ${\sqrt[3]{6 x}}^{2}$ as $\sqrt[3]{{(6 x)}^{2}}$ .

$y = \frac{\sqrt[3]{x^{6} + 6 C} \sqrt[3]{{(6 x)}^{2}}}{6 x}$

Step 13.2.9.2

Apply the product rule to $6 x$ .

$y = \frac{\sqrt[3]{x^{6} + 6 C} \sqrt[3]{6^{2} x^{2}}}{6 x}$

Step 13.2.9.3

Raise $6$ to the power of $2$ .

$y = \frac{\sqrt[3]{x^{6} + 6 C} \sqrt[3]{36 x^{2}}}{6 x}$

Step 13.2.10

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(x^{6} + 6 C) \cdot 36 x^{2}}}{6 x}$

Step 13.2.11

Reorder factors in $\frac{\sqrt[3]{(x^{6} + 6 C) \cdot 36 x^{2}}}{6 x}$ .

$y = \frac{\sqrt[3]{36 x^{2} (x^{6} + 6 C)}}{6 x}$

Step 14

Simplify the constant of integration.

$y = \frac{\sqrt[3]{36 x^{2} (x^{6} + C)}}{6 x}$