Calculus Examples

\frac{d y}{d x} - \frac{1}{x} y = 2 x

$\frac{d y}{d x} - \frac{1}{x} y = 2 x$

Step 1

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 1.1

Set up the integration.

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

Step 1.2

Integrate

$- \frac{1}{x}$ .

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Step 1.2.1

Since

$- 1$ is constant with respect to

$x$ , move

$- 1$ out of the integral.

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

Step 1.2.2

The integral of

$\frac{1}{x}$ with respect to

$x$ is

$\ln (| x |)$ .

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

Step 1.2.3

Simplify.

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

Step 1.3

Remove the constant of integration.

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

Step 1.4

Use the logarithmic power rule.

$e^{\ln (x^{- 1})}$

Step 1.5

Exponentiation and log are inverse functions.

$x^{- 1}$

Step 1.6

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x}$

Step 2

Multiply each term by the integrating factor

$\frac{1}{x}$ .

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Step 2.1

Multiply each term by

$\frac{1}{x}$ .

$\frac{1}{x} \frac{d y}{d x} + \frac{1}{x} (- \frac{1}{x} y) = \frac{1}{x} (2 x)$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine

$\frac{1}{x}$ and

$\frac{d y}{d x}$ .

$\frac{\frac{d y}{d x}}{x} + \frac{1}{x} (- \frac{1}{x} y) = \frac{1}{x} (2 x)$

Step 2.2.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{x} - \frac{1}{x} (\frac{1}{x} y) = \frac{1}{x} (2 x)$

Step 2.2.3

Combine

$\frac{1}{x}$ and

$y$ .

$\frac{\frac{d y}{d x}}{x} - \frac{1}{x} \cdot \frac{y}{x} = \frac{1}{x} (2 x)$

Step 2.2.4

Multiply

$- \frac{1}{x} \cdot \frac{y}{x}$ .

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Step 2.2.4.1

Multiply

$\frac{y}{x}$ by

$\frac{1}{x}$ .

$\frac{\frac{d y}{d x}}{x} - \frac{y}{x \cdot x} = \frac{1}{x} (2 x)$

Step 2.2.4.2

Raise

$x$ to the power of

$1$ .

$\frac{\frac{d y}{d x}}{x} - \frac{y}{x^{1} x} = \frac{1}{x} (2 x)$

Step 2.2.4.3

Raise

$x$ to the power of

$1$ .

$\frac{\frac{d y}{d x}}{x} - \frac{y}{x^{1} x^{1}} = \frac{1}{x} (2 x)$

Step 2.2.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{d y}{d x}}{x} - \frac{y}{x^{1 + 1}} = \frac{1}{x} (2 x)$

Step 2.2.4.5

Add

$1$ and

$1$ .

$\frac{\frac{d y}{d x}}{x} - \frac{y}{x^{2}} = \frac{1}{x} (2 x)$

Step 2.3

Rewrite using the commutative property of multiplication.

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

Step 2.4

Combine

$2$ and

$\frac{1}{x}$ .

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

Step 2.5

Cancel the common factor of

$x$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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

Step 3

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

$\frac{d}{d x} [\frac{1}{x} y] = 2$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x} y] d x = \int 2 d x$

Step 5

Integrate the left side.

$\frac{1}{x} y = \int 2 d x$

Step 6

Apply the constant rule.

$\frac{1}{x} y = 2 x + C$

Step 7

Solve for

$y$ .

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Step 7.1

Combine

$\frac{1}{x}$ and

$y$ .

$\frac{y}{x} = 2 x + C$

Step 7.2

Multiply both sides by

$x$ .

$\frac{y}{x} x = (2 x + C) x$

Step 7.3

Simplify.

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Step 7.3.1

Simplify the left side.

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Step 7.3.1.1

Cancel the common factor of

$x$ .

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Step 7.3.1.1.1

Cancel the common factor.

$\frac{y}{x} x = (2 x + C) x$

Step 7.3.1.1.2

Rewrite the expression.

$y = (2 x + C) x$

Step 7.3.2

Simplify the right side.

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Step 7.3.2.1

Simplify

$(2 x + C) x$ .

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Step 7.3.2.1.1

Apply the distributive property.

$y = 2 x \cdot x + C x$

Step 7.3.2.1.2

Multiply

$x$ by

$x$ by adding the exponents.

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Step 7.3.2.1.2.1

Move

$x$ .

$y = 2 (x \cdot x) + C x$

Step 7.3.2.1.2.2

Multiply

$x$ by

$x$ .

$y = 2 x^{2} + C x$

Step 7.3.2.1.3

Reorder

$2 x^{2}$ and

$C x$ .

$y = C x + 2 x^{2}$

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