Calculus Examples

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

Step 1

Rewrite the differential equation.

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

Step 2

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

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Step 2.1

Subtract $y$ from both sides of the equation.

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

Step 2.2

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

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

Step 2.3

Cancel the common factor of $x$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

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

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

Step 2.4.2.2

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

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

Step 2.4.2.3

Cancel the common factor.

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

Step 2.4.2.4

Rewrite the expression.

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

Step 2.4.2.5

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 3

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 3.1

Set up the integration.

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

Step 3.2

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

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Step 3.2.1

Split the fraction into multiple fractions.

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Simplify.

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

Step 3.3

Remove the constant of integration.

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

Step 3.4

Use the logarithmic power rule.

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

Step 3.5

Exponentiation and log are inverse functions.

$x^{- 1}$

Step 3.6

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

$\frac{1}{x}$

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

Move the negative in front of the fraction.

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

Step 4.2.3

Rewrite using the commutative property of multiplication.

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

Step 4.2.4

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

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

Step 4.2.5

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

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Step 4.2.5.1

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

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

Step 4.2.5.2

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

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

Step 4.2.5.3

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

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

Step 4.2.5.4

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

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

Step 4.2.5.5

Add $1$ and $1$ .

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

Step 4.3

Cancel the common factor of $x$ .

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factor.

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

Step 4.3.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Set up an integral on each side.

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

Step 7

Integrate the left side.

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

Step 8

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

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

Step 9

Solve for $y$ .

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Step 9.1

Simplify the left side.

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Step 9.1.1

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

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

Step 9.2

Multiply both sides by $x$ .

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

Step 9.3

Simplify.

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Step 9.3.1

Simplify the left side.

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Step 9.3.1.1

Cancel the common factor of $x$ .

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Step 9.3.1.1.1

Cancel the common factor.

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

Step 9.3.1.1.2

Rewrite the expression.

$y = (- \cos (x) + C) x$

Step 9.3.2

Simplify the right side.

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Step 9.3.2.1

Simplify $(- \cos (x) + C) x$ .

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Step 9.3.2.1.1

Apply the distributive property.

$y = - \cos (x) x + C x$

Step 9.3.2.1.2

Simplify the expression.

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Step 9.3.2.1.2.1

Reorder factors in $- \cos (x) x + C x$ .

$y = - x \cos (x) + C x$

Step 9.3.2.1.2.2

Reorder $- x \cos (x)$ and $C x$ .

$y = C x - x \cos (x)$