Calculus Examples

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

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

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

Step 1.2

Divide each term in $2 x \frac{d y}{d x} - y = x$ by $2 x$ .

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

Step 1.3

Cancel the common factor of $2$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

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

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

Step 1.5

Cancel the common factor of $x$ .

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Rewrite the expression.

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

Step 1.6

Factor $y$ out of $\frac{- y}{2 x}$ .

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

Step 1.7

Reorder $y$ and $\frac{- 1}{2 x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 1}{2 x}$ .

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $\frac{- 1}{2 x}$ .

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Step 2.2.1

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

Step 2.2.3

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$e^{- (\frac{1}{2} \int \frac{1}{x} d x)}$

Step 2.2.4

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

$e^{- \frac{1}{2} (\ln (| x |) + C)}$

Step 2.2.5

Simplify.

$e^{- \frac{1}{2} \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- \frac{1}{2} \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- \frac{1}{2}})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- \frac{1}{2}}$

Step 2.6

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

$\frac{1}{x^{\frac{1}{2}}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{\frac{1}{2}}}$ .

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

Step 3.2.5

Multiply $- \frac{1}{x^{\frac{1}{2}}} \cdot \frac{y}{2 x}$ .

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Step 3.2.5.1

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

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

Step 3.2.5.2

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

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Step 3.2.5.2.1

Move $x^{\frac{1}{2}}$ .

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

Step 3.2.5.2.2

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

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Step 3.2.5.2.2.1

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

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

Step 3.2.5.2.2.2

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

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

Step 3.2.5.2.3

Write $1$ as a fraction with a common denominator.

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

Step 3.2.5.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5.2.5

Add $1$ and $2$ .

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

Step 3.3

Combine.

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

Step 3.4

Multiply $1$ by $1$ .

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

Step 3.5

Move $2$ to the left of $x^{\frac{1}{2}}$ .

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 7.2

Apply basic rules of exponents.

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Step 7.2.1

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

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

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 7.2.2.3

Move the negative in front of the fraction.

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

Step 7.3

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

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

Step 7.4

Simplify the answer.

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Step 7.4.1

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

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

Step 7.4.2

Simplify.

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Step 7.4.2.1

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

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

Step 7.4.2.2

Cancel the common factor of $2$ .

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Step 7.4.2.2.1

Cancel the common factor.

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

Step 7.4.2.2.2

Rewrite the expression.

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

Step 7.4.2.3

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

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

Step 8

Solve for $y$ .

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

Rewrite the expression.

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

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $(x^{\frac{1}{2}} + C) x^{\frac{1}{2}}$ .

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Step 8.3.2.1.1

Apply the distributive property.

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

Step 8.3.2.1.2

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

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Step 8.3.2.1.2.1

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

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

Step 8.3.2.1.2.2

Combine the numerators over the common denominator.

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

Step 8.3.2.1.2.3

Add $1$ and $1$ .

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

Step 8.3.2.1.2.4

Divide $2$ by $2$ .

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

Step 8.3.2.1.3

Simplify $x^{1}$ .

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