Calculus Examples

$x^{2} \cdot \frac{d y}{d x} + 37 x y = 7 x - 47$

Step 1

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

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Step 1.1

Divide each term in $x^{2} \cdot \frac{d y}{d x} + 37 x y = 7 x - 47$ by $x^{2}$ .

$\frac{x^{2} \cdot \frac{d y}{d x}}{x^{2}} + \frac{37 x y}{x^{2}} = \frac{7 x}{x^{2}} + \frac{- 47}{x^{2}}$

Step 1.2

Cancel the common factor of $x^{2}$ .

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Step 1.2.1

Cancel the common factor.

$\frac{x^{2} \cdot \frac{d y}{d x}}{x^{2}} + \frac{37 x y}{x^{2}} = \frac{7 x}{x^{2}} + \frac{- 47}{x^{2}}$

Step 1.2.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} + \frac{37 x y}{x^{2}} = \frac{7 x}{x^{2}} + \frac{- 47}{x^{2}}$

Step 1.3

Cancel the common factor of $x$ and $x^{2}$ .

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Step 1.3.1

Factor $x$ out of $37 x y$ .

$\frac{d y}{d x} + \frac{x (37 y)}{x^{2}} = \frac{7 x}{x^{2}} + \frac{- 47}{x^{2}}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

$\frac{d y}{d x} + \frac{x (37 y)}{x \cdot x} = \frac{7 x}{x^{2}} + \frac{- 47}{x^{2}}$

Step 1.3.2.2

Cancel the common factor.

$\frac{d y}{d x} + \frac{x (37 y)}{x \cdot x} = \frac{7 x}{x^{2}} + \frac{- 47}{x^{2}}$

Step 1.3.2.3

Rewrite the expression.

$\frac{d y}{d x} + \frac{37 y}{x} = \frac{7 x}{x^{2}} + \frac{- 47}{x^{2}}$

Step 1.4

Cancel the common factor of $x$ and $x^{2}$ .

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Step 1.4.1

Factor $x$ out of $7 x$ .

$\frac{d y}{d x} + \frac{37 y}{x} = \frac{x \cdot 7}{x^{2}} + \frac{- 47}{x^{2}}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

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

$\frac{d y}{d x} + \frac{37 y}{x} = \frac{x \cdot 7}{x \cdot x} + \frac{- 47}{x^{2}}$

Step 1.4.2.2

Cancel the common factor.

$\frac{d y}{d x} + \frac{37 y}{x} = \frac{x \cdot 7}{x \cdot x} + \frac{- 47}{x^{2}}$

Step 1.4.2.3

Rewrite the expression.

$\frac{d y}{d x} + \frac{37 y}{x} = \frac{7}{x} + \frac{- 47}{x^{2}}$

Step 1.5

Factor $y$ out of $\frac{37 y}{x}$ .

$\frac{d y}{d x} + y \frac{37}{x} = \frac{7}{x} + \frac{- 47}{x^{2}}$

Step 1.6

Reorder $y$ and $\frac{37}{x}$ .

$\frac{d y}{d x} + \frac{37}{x} y = \frac{7}{x} + \frac{- 47}{x^{2}}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{37}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{37}{x} d x}$

Step 2.2

Integrate $\frac{37}{x}$ .

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Step 2.2.1

Since $37$ is constant with respect to $x$ , move $37$ out of the integral.

$e^{37 \int \frac{1}{x} d x}$

Step 2.2.2

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

$e^{37 (\ln (| x |) + C)}$

Step 2.2.3

Simplify.

$e^{37 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{37 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{37})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{37}$

Step 3

Multiply each term by the integrating factor $x^{37}$ .

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Step 3.1

Multiply each term by $x^{37}$ .

$x^{37} \frac{d y}{d x} + x^{37} (\frac{37}{x} y) = x^{37} \frac{7}{x} + x^{37} \frac{- 47}{x^{2}}$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{37}{x}$ and $y$ .

$x^{37} \frac{d y}{d x} + x^{37} \frac{37 y}{x} = x^{37} \frac{7}{x} + x^{37} \frac{- 47}{x^{2}}$

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

Factor $x$ out of $x^{37}$ .

$x^{37} \frac{d y}{d x} + x \cdot x^{36} \frac{37 y}{x} = x^{37} \frac{7}{x} + x^{37} \frac{- 47}{x^{2}}$

Step 3.2.2.2

Cancel the common factor.

$x^{37} \frac{d y}{d x} + x \cdot x^{36} \frac{37 y}{x} = x^{37} \frac{7}{x} + x^{37} \frac{- 47}{x^{2}}$

Step 3.2.2.3

Rewrite the expression.

$x^{37} \frac{d y}{d x} + x^{36} (37 y) = x^{37} \frac{7}{x} + x^{37} \frac{- 47}{x^{2}}$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$x^{37} \frac{d y}{d x} + 37 x^{36} y = x^{37} \frac{7}{x} + x^{37} \frac{- 47}{x^{2}}$

Step 3.3

Simplify each term.

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Step 3.3.1

Cancel the common factor of $x$ .

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Step 3.3.1.1

Factor $x$ out of $x^{37}$ .

$x^{37} \frac{d y}{d x} + 37 x^{36} y = x \cdot x^{36} \frac{7}{x} + x^{37} \frac{- 47}{x^{2}}$

Step 3.3.1.2

Cancel the common factor.

$x^{37} \frac{d y}{d x} + 37 x^{36} y = x \cdot x^{36} \frac{7}{x} + x^{37} \frac{- 47}{x^{2}}$

Step 3.3.1.3

Rewrite the expression.

$x^{37} \frac{d y}{d x} + 37 x^{36} y = x^{36} \cdot 7 + x^{37} \frac{- 47}{x^{2}}$

Step 3.3.2

Move $7$ to the left of $x^{36}$ .

$x^{37} \frac{d y}{d x} + 37 x^{36} y = 7 \cdot x^{36} + x^{37} \frac{- 47}{x^{2}}$

Step 3.3.3

Cancel the common factor of $x^{2}$ .

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Step 3.3.3.1

Factor $x^{2}$ out of $x^{37}$ .

$x^{37} \frac{d y}{d x} + 37 x^{36} y = 7 x^{36} + x^{2} x^{35} \frac{- 47}{x^{2}}$

Step 3.3.3.2

Cancel the common factor.

$x^{37} \frac{d y}{d x} + 37 x^{36} y = 7 x^{36} + x^{2} x^{35} \frac{- 47}{x^{2}}$

Step 3.3.3.3

Rewrite the expression.

$x^{37} \frac{d y}{d x} + 37 x^{36} y = 7 x^{36} + x^{35} \cdot - 47$

Step 3.3.4

Move $- 47$ to the left of $x^{35}$ .

$x^{37} \frac{d y}{d x} + 37 x^{36} y = 7 x^{36} - 47 x^{35}$

Step 4

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

$\frac{d}{d x} [x^{37} y] = 7 x^{36} - 47 x^{35}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{37} y] d x = \int 7 x^{36} - 47 x^{35} d x$

Step 6

Integrate the left side.

$x^{37} y = \int 7 x^{36} - 47 x^{35} d x$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$x^{37} y = \int 7 x^{36} d x + \int - 47 x^{35} d x$

Step 7.2

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

$x^{37} y = 7 \int x^{36} d x + \int - 47 x^{35} d x$

Step 7.3

By the Power Rule, the integral of $x^{36}$ with respect to $x$ is $\frac{1}{37} x^{37}$ .

$x^{37} y = 7 (\frac{1}{37} x^{37} + C) + \int - 47 x^{35} d x$

Step 7.4

Since $- 47$ is constant with respect to $x$ , move $- 47$ out of the integral.

$x^{37} y = 7 (\frac{1}{37} x^{37} + C) - 47 \int x^{35} d x$

Step 7.5

By the Power Rule, the integral of $x^{35}$ with respect to $x$ is $\frac{1}{36} x^{36}$ .

$x^{37} y = 7 (\frac{1}{37} x^{37} + C) - 47 (\frac{1}{36} x^{36} + C)$

Step 7.6

Simplify.

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Step 7.6.1

Simplify.

$x^{37} y = 7 (\frac{1}{37}) x^{37} - 47 (\frac{1}{36}) x^{36} + C$

Step 7.6.2

Simplify.

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Step 7.6.2.1

Combine $7$ and $\frac{1}{37}$ .

$x^{37} y = \frac{7}{37} x^{37} - 47 (\frac{1}{36}) x^{36} + C$

Step 7.6.2.2

Combine $- 47$ and $\frac{1}{36}$ .

$x^{37} y = \frac{7}{37} x^{37} + \frac{- 47}{36} x^{36} + C$

Step 7.6.2.3

Move the negative in front of the fraction.

$x^{37} y = \frac{7}{37} x^{37} - \frac{47}{36} x^{36} + C$

Step 8

Divide each term in $x^{37} y = \frac{7}{37} x^{37} - \frac{47}{36} x^{36} + C$ by $x^{37}$ and simplify.

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Step 8.1

Divide each term in $x^{37} y = \frac{7}{37} x^{37} - \frac{47}{36} x^{36} + C$ by $x^{37}$ .

$\frac{x^{37} y}{x^{37}} = \frac{\frac{7}{37} x^{37}}{x^{37}} + \frac{- \frac{47}{36} x^{36}}{x^{37}} + \frac{C}{x^{37}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $x^{37}$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{x^{37} y}{x^{37}} = \frac{\frac{7}{37} x^{37}}{x^{37}} + \frac{- \frac{47}{36} x^{36}}{x^{37}} + \frac{C}{x^{37}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{7}{37} x^{37}}{x^{37}} + \frac{- \frac{47}{36} x^{36}}{x^{37}} + \frac{C}{x^{37}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Cancel the common factor of $x^{37}$ .

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Step 8.3.1.1.1

Cancel the common factor.

$y = \frac{\frac{7}{37} x^{37}}{x^{37}} + \frac{- \frac{47}{36} x^{36}}{x^{37}} + \frac{C}{x^{37}}$

Step 8.3.1.1.2

Divide $\frac{7}{37}$ by $1$ .

$y = \frac{7}{37} + \frac{- \frac{47}{36} x^{36}}{x^{37}} + \frac{C}{x^{37}}$

Step 8.3.1.2

Cancel the common factor of $x^{36}$ and $x^{37}$ .

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Step 8.3.1.2.1

Factor $x^{36}$ out of $- \frac{47}{36} x^{36}$ .

$y = \frac{7}{37} + \frac{x^{36} (- \frac{47}{36})}{x^{37}} + \frac{C}{x^{37}}$

Step 8.3.1.2.2

Cancel the common factors.

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Step 8.3.1.2.2.1

Factor $x^{36}$ out of $x^{37}$ .

$y = \frac{7}{37} + \frac{x^{36} (- \frac{47}{36})}{x^{36} x} + \frac{C}{x^{37}}$

Step 8.3.1.2.2.2

Cancel the common factor.

$y = \frac{7}{37} + \frac{x^{36} (- \frac{47}{36})}{x^{36} x} + \frac{C}{x^{37}}$

Step 8.3.1.2.2.3

Rewrite the expression.

$y = \frac{7}{37} + \frac{- \frac{47}{36}}{x} + \frac{C}{x^{37}}$

Step 8.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{7}{37} - \frac{47}{36} \cdot \frac{1}{x} + \frac{C}{x^{37}}$

Step 8.3.1.4

Multiply $\frac{1}{x}$ by $\frac{47}{36}$ .

$y = \frac{7}{37} - \frac{47}{x \cdot 36} + \frac{C}{x^{37}}$

Step 8.3.1.5

Move $36$ to the left of $x$ .

$y = \frac{7}{37} - \frac{47}{36 x} + \frac{C}{x^{37}}$