Calculus Examples

$y - x \frac{d y}{d x} = x^{3}$

Step 1

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

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Step 1.1

Reorder terms.

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

Step 1.2

Divide each term in $- x \frac{d y}{d x} + y = x^{3}$ by $- x$ .

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

Step 1.3

Dividing two negative values results in a positive value.

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

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

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

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

Step 1.5

Cancel the common factor of $x^{3}$ and $x$ .

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Step 1.5.1

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

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

Step 1.5.2

Move the negative one from the denominator of $\frac{x^{2}}{- 1}$ .

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

Step 1.6

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

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

Step 1.7

Reorder $y$ and $\frac{1}{- x}$ .

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

Step 2

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 2.1

Set up the integration.

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor of $1$ and $- 1$ .

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Step 2.2.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 2.2.1.2

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Simplify.

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

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

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 1}$

Step 2.6

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

$\frac{1}{x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{1}{x}$ and $\frac{d y}{d x}$ .

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

Step 3.2.2

Cancel the common factor of $1$ and $- 1$ .

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Step 3.2.2.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.2.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

Multiply $\frac{y}{x}$ by $\frac{1}{x}$ .

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

Step 3.2.5.2

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

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

Step 3.2.5.3

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

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

Step 3.2.5.4

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

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

Step 3.2.5.5

Add $1$ and $1$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Cancel the common factor of $x$ .

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Step 3.4.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 3.4.2

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

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Rewrite $- (\frac{1}{2} x^{2} + C)$ as $- \frac{1}{2} x^{2} + C$ .

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

Step 8

Solve for $y$ .

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Multiply both sides by $x$ .

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

Step 8.4

Simplify.

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

Cancel the common factor of $x$ .

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Step 8.4.1.1.1

Cancel the common factor.

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

Step 8.4.1.1.2

Rewrite the expression.

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

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(- \frac{x^{2}}{2} + C) x$ .

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Step 8.4.2.1.1

Apply the distributive property.

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

Step 8.4.2.1.2

Multiply $- \frac{x^{2}}{2} x$ .

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Step 8.4.2.1.2.1

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

$y = - \frac{x \cdot x^{2}}{2} + C x$

Step 8.4.2.1.2.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 8.4.2.1.2.2.1

Multiply $x$ by $x^{2}$ .

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Step 8.4.2.1.2.2.1.1

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

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

Step 8.4.2.1.2.2.1.2

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

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

Step 8.4.2.1.2.2.2

Add $1$ and $2$ .

$y = - \frac{x^{3}}{2} + C x$