Calculus Examples

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

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

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

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

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

Step 1.4

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

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

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

Split the fraction into multiple fractions.

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

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 $\frac{1}{x}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

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 $\frac{1}{x}$ .

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

Step 3.2.5.2

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

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

Step 3.2.5.3

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

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

Step 3.2.5.4

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

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

Step 3.2.5.5

Add $1$ and $1$ .

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

Step 3.3

Combine.

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

Step 3.4

Multiply $x$ by $x$ .

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

Step 3.5

Multiply $\frac{1}{x^{2}}$ by $1$ .

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

Step 3.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.7

Combine.

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

Step 3.8

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

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Step 3.8.1

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

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

Step 3.8.2

Add $2$ and $2$ .

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

Step 3.9

Multiply $1$ by $1$ .

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

Step 4

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

$\frac{d}{d x} [\frac{1}{x} y] = \frac{1}{x^{4}}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x} y] d x = \int \frac{1}{x^{4}} d x$

Step 6

Integrate the left side.

$\frac{1}{x} y = \int \frac{1}{x^{4}} d x$

Step 7

Integrate the right side.

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Step 7.1

Apply basic rules of exponents.

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Step 7.1.1

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{x} y = \int {(x^{4})}^{- 1} d x$

Step 7.1.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 7.1.2.1

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

$\frac{1}{x} y = \int x^{4 \cdot - 1} d x$

Step 7.1.2.2

Multiply $4$ by $- 1$ .

$\frac{1}{x} y = \int x^{- 4} d x$

Step 7.2

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

$\frac{1}{x} y = - \frac{1}{3} x^{- 3} + C$

Step 7.3

Simplify the answer.

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Step 7.3.1

Rewrite $- \frac{1}{3} x^{- 3} + C$ as $- \frac{1}{3} \cdot \frac{1}{x^{3}} + C$ .

$\frac{1}{x} y = - \frac{1}{3} \cdot \frac{1}{x^{3}} + C$

Step 7.3.2

Simplify.

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Step 7.3.2.1

Multiply $\frac{1}{x^{3}}$ by $\frac{1}{3}$ .

$\frac{1}{x} y = - \frac{1}{x^{3} \cdot 3} + C$

Step 7.3.2.2

Move $3$ to the left of $x^{3}$ .

$\frac{1}{x} y = - \frac{1}{3 x^{3}} + C$

Step 8

Solve for $y$ .

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Step 8.1

Move all terms containing variables to the left side of the equation.

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Step 8.1.1

Add $\frac{1}{3 x^{3}}$ to both sides of the equation.

$\frac{1}{x} y + \frac{1}{3 x^{3}} = C$

Step 8.1.2

Subtract $C$ from both sides of the equation.

$\frac{1}{x} y + \frac{1}{3 x^{3}} - C = 0$

Step 8.1.3

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

$\frac{y}{x} + \frac{1}{3 x^{3}} - C = 0$

Step 8.2

Move all terms not containing $y$ to the right side of the equation.

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Step 8.2.1

Subtract $\frac{1}{3 x^{3}}$ from both sides of the equation.

$\frac{y}{x} - C = - \frac{1}{3 x^{3}}$

Step 8.2.2

Add $C$ to both sides of the equation.

$\frac{y}{x} = - \frac{1}{3 x^{3}} + C$

Step 8.3

Multiply both sides by $x$ .

$\frac{y}{x} x = (- \frac{1}{3 x^{3}} + C) x$

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.

$\frac{y}{x} x = (- \frac{1}{3 x^{3}} + C) x$

Step 8.4.1.1.2

Rewrite the expression.

$y = (- \frac{1}{3 x^{3}} + C) x$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(- \frac{1}{3 x^{3}} + C) x$ .

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Step 8.4.2.1.1

Apply the distributive property.

$y = - \frac{1}{3 x^{3}} x + C x$

Step 8.4.2.1.2

Cancel the common factor of $x$ .

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Step 8.4.2.1.2.1

Move the leading negative in $- \frac{1}{3 x^{3}}$ into the numerator.

$y = \frac{- 1}{3 x^{3}} x + C x$

Step 8.4.2.1.2.2

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

$y = \frac{- 1}{x (3 x^{2})} x + C x$

Step 8.4.2.1.2.3

Cancel the common factor.

$y = \frac{- 1}{x (3 x^{2})} x + C x$

Step 8.4.2.1.2.4

Rewrite the expression.

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

Step 8.4.2.1.3

Simplify the expression.

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Step 8.4.2.1.3.1

Move the negative in front of the fraction.

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

Step 8.4.2.1.3.2

Reorder $- \frac{1}{3 x^{2}}$ and $C x$ .

$y = C x - \frac{1}{3 x^{2}}$