Calculus Examples

$\frac{d y}{d x} = \frac{x f (x) - 16 (f x)}{x^{17}}$

Step 1

Rewrite the differential equation.

$\frac{d y}{d x} = \frac{x y - 16 (f x)}{x^{17}}$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Split the fraction $\frac{x y - 16 (f x)}{x^{17}}$ into two fractions.

$\frac{d y}{d x} = \frac{x y}{x^{17}} + \frac{- 16 (f x)}{x^{17}}$

Step 2.1.2

Subtract $\frac{x y}{x^{17}}$ from both sides of the equation.

$\frac{d y}{d x} - \frac{x y}{x^{17}} = \frac{- 16 (f x)}{x^{17}}$

Step 2.2

Rewrite the equation with isolated coefficients.

$\frac{d y}{d x} - \frac{x y}{x^{17}} = \frac{- 16 f x}{x^{17}}$

Step 2.3

Factor $y$ out of $- \frac{x y}{x^{17}}$ .

$\frac{d y}{d x} + y (- \frac{x}{x^{17}}) = \frac{- 16 f x}{x^{17}}$

Step 2.4

Reorder $y$ and $- \frac{x}{x^{17}}$ .

$\frac{d y}{d x} - \frac{x}{x^{17}} y = \frac{- 16 f x}{x^{17}}$

Step 3

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - \frac{x}{x^{17}}$ .

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Step 3.1

Set up the integration.

$e^{\int - \frac{x}{x^{17}} d x}$

Step 3.2

Integrate $- \frac{x}{x^{17}}$ .

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Step 3.2.1

Cancel the common factor of $x$ and $x^{17}$ .

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Step 3.2.1.1

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

$e^{\int - \frac{x^{1}}{x^{17}} d x}$

Step 3.2.1.2

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

$e^{\int - \frac{x \cdot 1}{x^{17}} d x}$

Step 3.2.1.3

Cancel the common factors.

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Step 3.2.1.3.1

Factor $x$ out of $x^{17}$ .

$e^{\int - \frac{x \cdot 1}{x \cdot x^{16}} d x}$

Step 3.2.1.3.2

Cancel the common factor.

$e^{\int - \frac{x \cdot 1}{x \cdot x^{16}} d x}$

Step 3.2.1.3.3

Rewrite the expression.

$e^{\int - \frac{1}{x^{16}} d x}$

Step 3.2.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$e^{- \int \frac{1}{x^{16}} d x}$

Step 3.2.3

Apply basic rules of exponents.

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Step 3.2.3.1

Move $x^{16}$ out of the denominator by raising it to the $- 1$ power.

$e^{- \int {(x^{16})}^{- 1} d x}$

Step 3.2.3.2

Multiply the exponents in ${(x^{16})}^{- 1}$ .

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Step 3.2.3.2.1

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

$e^{- \int x^{16 \cdot - 1} d x}$

Step 3.2.3.2.2

Multiply $16$ by $- 1$ .

$e^{- \int x^{- 16} d x}$

Step 3.2.4

By the Power Rule, the integral of $x^{- 16}$ with respect to $x$ is $- \frac{1}{15} x^{- 15}$ .

$e^{- (- \frac{1}{15} x^{- 15} + C)}$

Step 3.2.5

Simplify the answer.

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Step 3.2.5.1

Simplify.

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Step 3.2.5.1.1

Combine $x^{- 15}$ and $\frac{1}{15}$ .

$e^{- (- \frac{x^{- 15}}{15} + C)}$

Step 3.2.5.1.2

Move $x^{- 15}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$e^{- (- \frac{1}{15 x^{15}} + C)}$

Step 3.2.5.2

Simplify.

$e^{- - \frac{1}{15 x^{15}} + C}$

Step 3.2.5.3

Simplify.

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Step 3.2.5.3.1

Multiply $- 1$ by $- 1$ .

$e^{1 \frac{1}{15 x^{15}} + C}$

Step 3.2.5.3.2

Multiply $\frac{1}{15 x^{15}}$ by $1$ .

$e^{\frac{1}{15 x^{15}} + C}$

Step 3.3

Remove the constant of integration.

$e^{\frac{1}{15 x^{15}}}$

Step 4

Multiply each term by the integrating factor $e^{\frac{1}{15 x^{15}}}$ .

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Step 4.1

Multiply each term by $e^{\frac{1}{15 x^{15}}}$ .

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} + e^{\frac{1}{15 x^{15}}} (- \frac{x}{x^{17}} y) = e^{\frac{1}{15 x^{15}}} \frac{- 16 f x}{x^{17}}$

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite using the commutative property of multiplication.

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - e^{\frac{1}{15 x^{15}}} (\frac{x}{x^{17}} y) = e^{\frac{1}{15 x^{15}}} \frac{- 16 f x}{x^{17}}$

Step 4.2.2

Cancel the common factor of $x$ and $x^{17}$ .

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Step 4.2.2.1

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

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - e^{\frac{1}{15 x^{15}}} (\frac{x^{1}}{x^{17}} y) = e^{\frac{1}{15 x^{15}}} \frac{- 16 f x}{x^{17}}$

Step 4.2.2.2

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

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - e^{\frac{1}{15 x^{15}}} (\frac{x \cdot 1}{x^{17}} y) = e^{\frac{1}{15 x^{15}}} \frac{- 16 f x}{x^{17}}$

Step 4.2.2.3

Cancel the common factors.

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Step 4.2.2.3.1

Factor $x$ out of $x^{17}$ .

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - e^{\frac{1}{15 x^{15}}} (\frac{x \cdot 1}{x \cdot x^{16}} y) = e^{\frac{1}{15 x^{15}}} \frac{- 16 f x}{x^{17}}$

Step 4.2.2.3.2

Cancel the common factor.

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - e^{\frac{1}{15 x^{15}}} (\frac{x \cdot 1}{x \cdot x^{16}} y) = e^{\frac{1}{15 x^{15}}} \frac{- 16 f x}{x^{17}}$

Step 4.2.2.3.3

Rewrite the expression.

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - e^{\frac{1}{15 x^{15}}} (\frac{1}{x^{16}} y) = e^{\frac{1}{15 x^{15}}} \frac{- 16 f x}{x^{17}}$

Step 4.2.3

Combine $\frac{1}{x^{16}}$ and $y$ .

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - e^{\frac{1}{15 x^{15}}} \frac{y}{x^{16}} = e^{\frac{1}{15 x^{15}}} \frac{- 16 f x}{x^{17}}$

Step 4.2.4

Combine $\frac{y}{x^{16}}$ and $e^{\frac{1}{15 x^{15}}}$ .

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - \frac{y e^{\frac{1}{15 x^{15}}}}{x^{16}} = e^{\frac{1}{15 x^{15}}} \frac{- 16 f x}{x^{17}}$

Step 4.3

Cancel the common factor of $x$ and $x^{17}$ .

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Step 4.3.1

Factor $x$ out of $- 16 f x$ .

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - \frac{y e^{\frac{1}{15 x^{15}}}}{x^{16}} = e^{\frac{1}{15 x^{15}}} \frac{x (- 16 f)}{x^{17}}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $x$ out of $x^{17}$ .

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - \frac{y e^{\frac{1}{15 x^{15}}}}{x^{16}} = e^{\frac{1}{15 x^{15}}} \frac{x (- 16 f)}{x \cdot x^{16}}$

Step 4.3.2.2

Cancel the common factor.

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - \frac{y e^{\frac{1}{15 x^{15}}}}{x^{16}} = e^{\frac{1}{15 x^{15}}} \frac{x (- 16 f)}{x \cdot x^{16}}$

Step 4.3.2.3

Rewrite the expression.

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - \frac{y e^{\frac{1}{15 x^{15}}}}{x^{16}} = e^{\frac{1}{15 x^{15}}} \frac{- 16 f}{x^{16}}$

Step 4.4

Move the negative in front of the fraction.

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - \frac{y e^{\frac{1}{15 x^{15}}}}{x^{16}} = e^{\frac{1}{15 x^{15}}} (- \frac{16 f}{x^{16}})$

Step 4.5

Rewrite using the commutative property of multiplication.

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - \frac{y e^{\frac{1}{15 x^{15}}}}{x^{16}} = - e^{\frac{1}{15 x^{15}}} \frac{16 f}{x^{16}}$

Step 4.6

Combine $\frac{16 f}{x^{16}}$ and $e^{\frac{1}{15 x^{15}}}$ .

$e^{\frac{1}{15 x^{15}}} \frac{d y}{d x} - \frac{y e^{\frac{1}{15 x^{15}}}}{x^{16}} = - \frac{16 f e^{\frac{1}{15 x^{15}}}}{x^{16}}$

Step 5

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

$\frac{d}{d x} [e^{\frac{1}{15 x^{15}}} y] = - \frac{16 f e^{\frac{1}{15 x^{15}}}}{x^{16}}$

Step 6

Set up an integral on each side.

$\int \frac{d}{d x} [e^{\frac{1}{15 x^{15}}} y] d x = \int - \frac{16 f e^{\frac{1}{15 x^{15}}}}{x^{16}} d x$

Step 7

Integrate the left side.

$e^{\frac{1}{15 x^{15}}} y = \int - \frac{16 f e^{\frac{1}{15 x^{15}}}}{x^{16}} d x$

Step 8

Integrate the right side.

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Step 8.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$e^{\frac{1}{15 x^{15}}} y = - \int \frac{16 f e^{\frac{1}{15 x^{15}}}}{x^{16}} d x$

Step 8.2

Since $16 f$ is constant with respect to $x$ , move $16 f$ out of the integral.

$e^{\frac{1}{15 x^{15}}} y = - (16 f \int \frac{e^{\frac{1}{15 x^{15}}}}{x^{16}} d x)$

Step 8.3

Multiply $16$ by $- 1$ .

$e^{\frac{1}{15 x^{15}}} y = - 16 f \int \frac{e^{\frac{1}{15 x^{15}}}}{x^{16}} d x$

Step 8.4

Let $u = \frac{1}{15 x^{15}}$ . Then $d u = - \frac{1}{x^{16}} d x$ , so $- d u = \frac{1}{x^{16}} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.4.1

Let $u = \frac{1}{15 x^{15}}$ . Find $\frac{d u}{d x}$ .

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Step 8.4.1.1

Differentiate $\frac{1}{15 x^{15}}$ .

$\frac{d}{d x} [\frac{1}{15 x^{15}}]$

Step 8.4.1.2

Since $\frac{1}{15}$ is constant with respect to $x$ , the derivative of $\frac{1}{15 x^{15}}$ with respect to $x$ is $\frac{1}{15} \frac{d}{d x} [\frac{1}{x^{15}}]$ .

$\frac{1}{15} \frac{d}{d x} [\frac{1}{x^{15}}]$

Step 8.4.1.3

Apply basic rules of exponents.

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Step 8.4.1.3.1

Rewrite $\frac{1}{x^{15}}$ as ${(x^{15})}^{- 1}$ .

$\frac{1}{15} \frac{d}{d x} [{(x^{15})}^{- 1}]$

Step 8.4.1.3.2

Multiply the exponents in ${(x^{15})}^{- 1}$ .

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Step 8.4.1.3.2.1

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

$\frac{1}{15} \frac{d}{d x} [x^{15 \cdot - 1}]$

Step 8.4.1.3.2.2

Multiply $15$ by $- 1$ .

$\frac{1}{15} \frac{d}{d x} [x^{- 15}]$

Step 8.4.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 15$ .

$\frac{1}{15} (- 15 x^{- 16})$

Step 8.4.1.5

Simplify terms.

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Step 8.4.1.5.1

Combine $- 15$ and $\frac{1}{15}$ .

$\frac{- 15}{15} x^{- 16}$

Step 8.4.1.5.2

Combine $\frac{- 15}{15}$ and $x^{- 16}$ .

$\frac{- 15 x^{- 16}}{15}$

Step 8.4.1.5.3

Move $x^{- 16}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 15}{15 x^{16}}$

Step 8.4.1.5.4

Cancel the common factor of $- 15$ and $15$ .

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Step 8.4.1.5.4.1

Factor $15$ out of $- 15$ .

$\frac{15 \cdot - 1}{15 x^{16}}$

Step 8.4.1.5.4.2

Cancel the common factors.

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Step 8.4.1.5.4.2.1

Factor $15$ out of $15 x^{16}$ .

$\frac{15 \cdot - 1}{15 (x^{16})}$

Step 8.4.1.5.4.2.2

Cancel the common factor.

$\frac{15 \cdot - 1}{15 x^{16}}$

Step 8.4.1.5.4.2.3

Rewrite the expression.

$\frac{- 1}{x^{16}}$

Step 8.4.1.5.5

Move the negative in front of the fraction.

$- \frac{1}{x^{16}}$

Step 8.4.2

Rewrite the problem using $u$ and $d u$ .

$e^{\frac{1}{15 x^{15}}} y = - 16 f \int - e^{u} d u$

Step 8.5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$e^{\frac{1}{15 x^{15}}} y = - 16 f (- \int e^{u} d u)$

Step 8.6

Multiply $- 1$ by $- 16$ .

$e^{\frac{1}{15 x^{15}}} y = 16 f \int e^{u} d u$

Step 8.7

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{\frac{1}{15 x^{15}}} y = 16 f (e^{u} + C)$

Step 8.8

Simplify.

$e^{\frac{1}{15 x^{15}}} y = 16 f e^{u} + C$

Step 8.9

Replace all occurrences of $u$ with $\frac{1}{15 x^{15}}$ .

$e^{\frac{1}{15 x^{15}}} y = 16 f e^{\frac{1}{15 x^{15}}} + C$

Step 9

Divide each term in $e^{\frac{1}{15 x^{15}}} y = 16 f e^{\frac{1}{15 x^{15}}} + C$ by $e^{\frac{1}{15 x^{15}}}$ and simplify.

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Step 9.1

Divide each term in $e^{\frac{1}{15 x^{15}}} y = 16 f e^{\frac{1}{15 x^{15}}} + C$ by $e^{\frac{1}{15 x^{15}}}$ .

$\frac{e^{\frac{1}{15 x^{15}}} y}{e^{\frac{1}{15 x^{15}}}} = \frac{16 f e^{\frac{1}{15 x^{15}}}}{e^{\frac{1}{15 x^{15}}}} + \frac{C}{e^{\frac{1}{15 x^{15}}}}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $e^{\frac{1}{15 x^{15}}}$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{e^{\frac{1}{15 x^{15}}} y}{e^{\frac{1}{15 x^{15}}}} = \frac{16 f e^{\frac{1}{15 x^{15}}}}{e^{\frac{1}{15 x^{15}}}} + \frac{C}{e^{\frac{1}{15 x^{15}}}}$

Step 9.2.1.2

Divide $y$ by $1$ .

$y = \frac{16 f e^{\frac{1}{15 x^{15}}}}{e^{\frac{1}{15 x^{15}}}} + \frac{C}{e^{\frac{1}{15 x^{15}}}}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Cancel the common factor of $e^{\frac{1}{15 x^{15}}}$ .

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Step 9.3.1.1

Cancel the common factor.

$y = \frac{16 f e^{\frac{1}{15 x^{15}}}}{e^{\frac{1}{15 x^{15}}}} + \frac{C}{e^{\frac{1}{15 x^{15}}}}$

Step 9.3.1.2

Divide $16 f$ by $1$ .

$y = 16 f + \frac{C}{e^{\frac{1}{15 x^{15}}}}$