Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

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

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Step 2.2.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.2.2

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

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

Step 2.2.3

Simplify.

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

Step 2.3

Remove the constant of integration.

$e^{2 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{2})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{2}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 4

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

$\frac{d}{d x} [x^{2} y] = x \ln (x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} y] d x = \int x \ln (x) d x$

Step 6

Integrate the left side.

$x^{2} y = \int x \ln (x) d x$

Step 7

Integrate the right side.

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Step 7.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x$ .

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

Combine $\ln (x)$ and $\frac{x^{2}}{2}$ .

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

Step 7.3

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

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

Step 7.4

Simplify.

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Step 7.4.1

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

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

Step 7.4.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 7.4.2.1

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

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

Step 7.4.2.2

Cancel the common factors.

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Step 7.4.2.2.1

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

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

Step 7.4.2.2.2

Factor $x$ out of $x^{1}$ .

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

Step 7.4.2.2.3

Cancel the common factor.

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

Step 7.4.2.2.4

Rewrite the expression.

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

Step 7.4.2.2.5

Divide $x$ by $1$ .

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

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

Step 7.5

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

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

Step 7.6

Simplify the answer.

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Step 7.6.1

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

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

Step 7.6.2

Simplify.

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Step 7.6.2.1

Combine $\frac{1}{2}$ and $\ln (x)$ .

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

Step 7.6.2.2

Combine $\frac{\ln (x)}{2}$ and $x^{2}$ .

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

Step 7.6.2.3

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

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

Step 7.6.2.4

Multiply $2$ by $2$ .

$x^{2} y = \frac{\ln (x) x^{2}}{2} - \frac{1}{4} x^{2} + C$

Step 7.6.3

Combine $x^{2}$ and $\frac{1}{4}$ .

$x^{2} y = \frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{4} + C$

Step 7.6.4

Reorder terms.

$x^{2} y = \frac{1}{2} \ln (x) x^{2} - \frac{1}{4} x^{2} + C$

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

Combine $x^{2}$ and $\frac{1}{4}$ .

$x^{2} y = \frac{1}{2} \cdot (\ln (x) x^{2}) - \frac{x^{2}}{4} + C$

Step 8.1.2

Remove parentheses.

$x^{2} y = \frac{1}{2} \cdot \ln (x) x^{2} - \frac{x^{2}}{4} + C$

Step 8.2

Divide each term in $x^{2} y = \frac{1}{2} \cdot (\ln (x) x^{2}) - \frac{x^{2}}{4} + C$ by $x^{2}$ and simplify.

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Step 8.2.1

Divide each term in $x^{2} y = \frac{1}{2} \cdot (\ln (x) x^{2}) - \frac{x^{2}}{4} + C$ by $x^{2}$ .

$\frac{x^{2} y}{x^{2}} = \frac{\frac{1}{2} \cdot (\ln (x) x^{2})}{x^{2}} + \frac{- \frac{x^{2}}{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{x^{2} y}{x^{2}} = \frac{\frac{1}{2} \cdot (\ln (x) x^{2})}{x^{2}} + \frac{- \frac{x^{2}}{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{2} \cdot (\ln (x) x^{2})}{x^{2}} + \frac{- \frac{x^{2}}{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Simplify each term.

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Step 8.2.3.1.1

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

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Step 8.2.3.1.1.1

Cancel the common factor.

$y = \frac{\frac{1}{2} \cdot (\ln (x) x^{2})}{x^{2}} + \frac{- \frac{x^{2}}{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.3.1.1.2

Divide $\frac{1}{2} \cdot (\ln (x))$ by $1$ .

$y = \frac{1}{2} \cdot (\ln (x)) + \frac{- \frac{x^{2}}{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.3.1.2

Simplify $\frac{1}{2} (\ln (x))$ by moving $\frac{1}{2}$ inside the logarithm.

$y = \ln (x^{\frac{1}{2}}) + \frac{- \frac{x^{2}}{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$y = \ln (x^{\frac{1}{2}}) - \frac{x^{2}}{4} \cdot \frac{1}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.3.1.4

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

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Step 8.2.3.1.4.1

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

$y = \ln (x^{\frac{1}{2}}) + \frac{- x^{2}}{4} \cdot \frac{1}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.3.1.4.2

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

$y = \ln (x^{\frac{1}{2}}) + \frac{x^{2} \cdot - 1}{4} \cdot \frac{1}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.3.1.4.3

Cancel the common factor.

$y = \ln (x^{\frac{1}{2}}) + \frac{x^{2} \cdot - 1}{4} \cdot \frac{1}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.3.1.4.4

Rewrite the expression.

$y = \ln (x^{\frac{1}{2}}) + \frac{- 1}{4} + \frac{C}{x^{2}}$

Step 8.2.3.1.5

Move the negative in front of the fraction.

$y = \ln (x^{\frac{1}{2}}) - \frac{1}{4} + \frac{C}{x^{2}}$