Calculus Examples

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

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 $- \frac{y}{x}$ .

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

Step 1.2

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

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

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

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

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

Step 2.2.2

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

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

Step 2.2.3

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 $\frac{1}{x}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.3

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

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

Step 3.2.4

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

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Step 3.2.4.1

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

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

Step 3.2.4.2

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

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

Step 3.2.4.3

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

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

Step 3.2.4.4

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

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

Step 3.2.4.5

Add $1$ and $1$ .

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 4

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

$\frac{d}{d x} [\frac{1}{x} y] = x \sin (2 x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x} y] d x = \int x \sin (2 x) d x$

Step 6

Integrate the left side.

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

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.3

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

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

Step 7.4

Simplify.

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Step 7.4.1

Multiply $- 1$ by $- 1$ .

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

Step 7.4.2

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

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

Step 7.5

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.5.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 7.5.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 7.5.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 7.5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 7.5.1.4

Multiply $2$ by $1$ .

$2$

Step 7.5.2

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{x} y = - \frac{x \cos (2 x)}{2} + \frac{1}{2} \int \cos (u) \frac{1}{2} d u$

Step 7.6

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{1}{x} y = - \frac{x \cos (2 x)}{2} + \frac{1}{2} \int \frac{\cos (u)}{2} d u$

Step 7.7

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

$\frac{1}{x} y = - \frac{x \cos (2 x)}{2} + \frac{1}{2} (\frac{1}{2} \int \cos (u) d u)$

Step 7.8

Simplify.

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Step 7.8.1

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

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

Step 7.8.2

Multiply $2$ by $2$ .

$\frac{1}{x} y = - \frac{x \cos (2 x)}{2} + \frac{1}{4} \int \cos (u) d u$

Step 7.9

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

$\frac{1}{x} y = - \frac{x \cos (2 x)}{2} + \frac{1}{4} (\sin (u) + C)$

Step 7.10

Simplify.

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Step 7.10.1

Rewrite $- \frac{x \cos (2 x)}{2} + \frac{1}{4} (\sin (u) + C)$ as $- \frac{1}{2} x \cos (2 x) + \frac{1}{4} \sin (u) + C$ .

$\frac{1}{x} y = - \frac{1}{2} x \cos (2 x) + \frac{1}{4} \sin (u) + C$

Step 7.10.2

Simplify.

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Step 7.10.2.1

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

$\frac{1}{x} y = - \frac{x}{2} \cos (2 x) + \frac{1}{4} \sin (u) + C$

Step 7.10.2.2

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

$\frac{1}{x} y = - \frac{\cos (2 x) x}{2} + \frac{1}{4} \sin (u) + C$

Step 7.11

Replace all occurrences of $u$ with $2 x$ .

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

Step 7.12

Reorder factors in $\frac{\cos (2 x) x}{2}$ .

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

Step 8

Solve for $y$ .

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Step 8.1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Apply the distributive property.

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

Step 8.3.1.2

Multiply $x$ by $1$ .

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

Step 8.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 8.3.1.4

Apply the distributive property.

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

Step 8.3.1.5

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

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

Step 8.3.1.6

Cancel the common factor of $2$ .

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Step 8.3.1.6.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 8.3.1.6.2

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

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

Step 8.3.1.6.3

Cancel the common factor.

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

Step 8.3.1.6.4

Rewrite the expression.

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

Step 8.3.1.7

Multiply $- 1$ by $- 1$ .

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

Step 8.3.1.8

Multiply $x$ by $1$ .

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

Step 8.3.1.9

Apply the sine double-angle identity.

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

Step 8.3.1.10

Cancel the common factor of $2$ .

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Step 8.3.1.10.1

Factor $2$ out of $4$ .

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

Step 8.3.1.10.2

Factor $2$ out of $2 \sin (x) \cos (x)$ .

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

Step 8.3.1.10.3

Cancel the common factor.

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

Step 8.3.1.10.4

Rewrite the expression.

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

Step 8.3.1.11

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

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

Step 8.3.1.12

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

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

Step 8.4

Solve the equation for $y$ .

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Step 8.4.1

Multiply both sides by $x$ .

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

Step 8.4.2

Simplify.

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Step 8.4.2.1

Simplify the left side.

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Step 8.4.2.1.1

Cancel the common factor of $x$ .

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Step 8.4.2.1.1.1

Cancel the common factor.

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

Step 8.4.2.1.1.2

Rewrite the expression.

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

Step 8.4.2.2

Simplify the right side.

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Step 8.4.2.2.1

Simplify $(- \frac{x}{2} + x \sin^{2} (x) + \frac{\sin (x) \cos (x)}{2} + C) x$ .

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Step 8.4.2.2.1.1

Apply the distributive property.

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

Step 8.4.2.2.1.2

Simplify.

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Step 8.4.2.2.1.2.1

Multiply $- \frac{x}{2} x$ .

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Step 8.4.2.2.1.2.1.1

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

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

Step 8.4.2.2.1.2.1.2

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

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

Step 8.4.2.2.1.2.1.3

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

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

Step 8.4.2.2.1.2.1.4

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

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

Step 8.4.2.2.1.2.1.5

Add $1$ and $1$ .

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

Step 8.4.2.2.1.2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 8.4.2.2.1.2.2.1

Move $x$ .

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

Step 8.4.2.2.1.2.2.2

Multiply $x$ by $x$ .

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

Step 8.4.2.2.1.2.3

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

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

Step 8.4.2.2.1.3

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

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

Step 8.4.2.2.1.4

Move $\frac{x \sin (x) \cos (x)}{2}$ .

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

Step 8.4.2.2.1.5

Move $x^{2} \sin^{2} (x)$ .

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