Calculus Examples

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

Step 1

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

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Step 1.1

Divide each term in $x \frac{d y}{d x} + 2 y = x \sin (x)$ by $x$ .

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Divide $\sin (x)$ by $1$ .

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

Step 1.4

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

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

Step 1.5

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

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

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 $x^{2}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

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 $u = x^{2}$ and $d v = \sin (x)$ .

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

Step 7.2

Multiply $2$ by $- 1$ .

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

Step 7.3

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

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

Step 7.4

Multiply $- 2$ by $- 1$ .

$x^{2} y = x^{2} (- \cos (x)) + 2 \int \cos (x) x d x$

Step 7.5

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

$x^{2} y = x^{2} (- \cos (x)) + 2 (x \sin (x) - \int \sin (x) d x)$

Step 7.6

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

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

Step 7.7

Simplify the answer.

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Step 7.7.1

Rewrite $x^{2} (- \cos (x)) + 2 (x \sin (x) - (- \cos (x) + C))$ as $- x^{2} \cos (x) + 2 (x \sin (x) - - \cos (x)) + C$ .

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

Step 7.7.2

Simplify.

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Step 7.7.2.1

Multiply $- 1$ by $- 1$ .

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

Step 7.7.2.2

Multiply $\cos (x)$ by $1$ .

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

Step 8

Divide each term in $x^{2} y = - x^{2} \cos (x) + 2 (x \sin (x) + \cos (x)) + C$ by $x^{2}$ and simplify.

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Step 8.1

Divide each term in $x^{2} y = - x^{2} \cos (x) + 2 (x \sin (x) + \cos (x)) + C$ by $x^{2}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify terms.

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Step 8.3.1.1

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

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

Divide $- \cos (x)$ by $1$ .

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

Step 8.3.1.2

Combine the numerators over the common denominator.

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

Step 8.3.2

Apply the distributive property.

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

Step 8.3.3

To write $- \cos (x)$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

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

Step 8.3.4

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

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

Step 8.3.5

Combine the numerators over the common denominator.

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

Step 8.3.6

Factor $- 1$ out of $- \cos (x) x^{2}$ .

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

Step 8.3.7

Factor $- 1$ out of $2 x \sin (x)$ .

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

Step 8.3.8

Factor $- 1$ out of $- (\cos (x) x^{2}) - (- 2 x \sin (x))$ .

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

Step 8.3.9

Factor $- 1$ out of $2 \cos (x)$ .

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

Step 8.3.10

Factor $- 1$ out of $- (\cos (x) x^{2} - 2 x \sin (x)) - (- 2 \cos (x))$ .

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

Step 8.3.11

Factor $- 1$ out of $C$ .

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

Step 8.3.12

Factor $- 1$ out of $- (\cos (x) x^{2} - 2 x \sin (x) - 2 \cos (x)) - 1 (- C)$ .

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

Step 8.3.13

Simplify the expression.

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Step 8.3.13.1

Rewrite $- (\cos (x) x^{2} - 2 x \sin (x) - 2 \cos (x) - C)$ as $- 1 (\cos (x) x^{2} - 2 x \sin (x) - 2 \cos (x) - C)$ .

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

Step 8.3.13.2

Move the negative in front of the fraction.

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

Step 8.3.13.3

Reorder factors in $- \frac{\cos (x) x^{2} - 2 x \sin (x) - 2 \cos (x) - C}{x^{2}}$ .

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