Calculus Examples

$x \frac{d y}{d x} + 1 = x + y$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Subtract $y$ from both sides of the equation.

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

Step 1.1.2

Subtract $1$ from both sides of the equation.

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

Step 1.2

Divide each term in $x \frac{d y}{d x} - y = x - 1$ by $x$ .

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

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

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

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

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

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

Step 1.6

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

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

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 $\frac{1}{x}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

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 $\frac{1}{x}$ .

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

Step 3.2.5.2

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

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

Step 3.2.5.3

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

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

Step 3.2.5.4

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

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

Step 3.2.5.5

Add $1$ and $1$ .

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

Step 3.3

Simplify each term.

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Step 3.3.1

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

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

Step 3.3.2

Move the negative in front of the fraction.

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

Step 3.3.3

Rewrite using the commutative property of multiplication.

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

Step 3.3.4

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

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Step 3.3.4.1

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

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

Step 3.3.4.2

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

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

Step 3.3.4.3

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

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

Step 3.3.4.4

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

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

Step 3.3.4.5

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Apply basic rules of exponents.

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Step 7.4.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 7.4.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 7.4.2.1

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

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

Step 7.4.2.2

Multiply $2$ by $- 1$ .

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

Step 7.5

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

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

Step 7.6

Simplify.

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Step 7.6.1

Simplify.

$\frac{1}{x} y = \ln (| x |) - - \frac{1}{x} + C$

Step 7.6.2

Simplify.

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Step 7.6.2.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{x} y = \ln (| x |) + 1 \frac{1}{x} + C$

Step 7.6.2.2

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

$\frac{1}{x} y = \ln (| x |) + \frac{1}{x} + C$

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

Subtract $\ln (| x |)$ from both sides of the equation.

$\frac{1}{x} y - \ln (| x |) = \frac{1}{x} + C$

Step 8.1.2

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

$\frac{1}{x} y - \ln (| x |) - \frac{1}{x} = C$

Step 8.1.3

Subtract $C$ from both sides of the equation.

$\frac{1}{x} y - \ln (| x |) - \frac{1}{x} - C = 0$

Step 8.1.4

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

$\frac{y}{x} - \ln (| x |) - \frac{1}{x} - C = 0$

Step 8.2

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

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Step 8.2.1

Add $\ln (| x |)$ to both sides of the equation.

$\frac{y}{x} - \frac{1}{x} - C = \ln (| x |)$

Step 8.2.2

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

$\frac{y}{x} - C = \ln (| x |) + \frac{1}{x}$

Step 8.2.3

Add $C$ to both sides of the equation.

$\frac{y}{x} = \ln (| x |) + \frac{1}{x} + C$

Step 8.3

Multiply both sides by $x$ .

$\frac{y}{x} x = (\ln (| x |) + \frac{1}{x} + C) x$

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.

$\frac{y}{x} x = (\ln (| x |) + \frac{1}{x} + C) x$

Step 8.4.1.1.2

Rewrite the expression.

$y = (\ln (| x |) + \frac{1}{x} + C) x$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(\ln (| x |) + \frac{1}{x} + C) x$ .

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Step 8.4.2.1.1

Apply the distributive property.

$y = \ln (| x |) x + \frac{1}{x} x + C x$

Step 8.4.2.1.2

Cancel the common factor of $x$ .

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Step 8.4.2.1.2.1

Cancel the common factor.

$y = \ln (| x |) x + \frac{1}{x} x + C x$

Step 8.4.2.1.2.2

Rewrite the expression.

$y = \ln (| x |) x + 1 + C x$

Step 8.4.2.1.3

Simplify the expression.

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Step 8.4.2.1.3.1

Reorder factors in $\ln (| x |) x + 1 + C x$ .

$y = x \ln (| x |) + 1 + C x$

Step 8.4.2.1.3.2

Move $1$ .

$y = x \ln (| x |) + C x + 1$

Step 8.4.2.1.3.3

Reorder $x \ln (| x |)$ and $C x$ .

$y = C x + x \ln (| x |) + 1$