Calculus Examples

$x \frac{d y}{d x} + 3 (y + x^{2}) = \frac{\sin (x)}{x}$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Apply the distributive property.

$x \frac{d y}{d x} + 3 y + 3 x^{2} = \frac{\sin (x)}{x}$

Step 1.1.2

Subtract $3 x^{2}$ from both sides of the equation.

$x \frac{d y}{d x} + 3 y = \frac{\sin (x)}{x} - 3 x^{2}$

Step 1.1.3

Reorder terms.

$x \frac{d y}{d x} + 3 y = - 3 x^{2} + \frac{\sin (x)}{x}$

Step 1.2

Divide each term in $x \frac{d y}{d x} + 3 y = - 3 x^{2} + \frac{\sin (x)}{x}$ by $x$ .

$\frac{x \frac{d y}{d x}}{x} + \frac{3 y}{x} = \frac{- 3 x^{2}}{x} + \frac{\frac{\sin (x)}{x}}{x}$

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} + \frac{3 y}{x} = \frac{- 3 x^{2}}{x} + \frac{\frac{\sin (x)}{x}}{x}$

Step 1.3.2

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

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{- 3 x^{2}}{x} + \frac{\frac{\sin (x)}{x}}{x}$

Step 1.4

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

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Step 1.4.1

Factor $x$ out of $- 3 x^{2}$ .

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{x (- 3 x)}{x} + \frac{\frac{\sin (x)}{x}}{x}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

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

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{x (- 3 x)}{x^{1}} + \frac{\frac{\sin (x)}{x}}{x}$

Step 1.4.2.2

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

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{x (- 3 x)}{x \cdot 1} + \frac{\frac{\sin (x)}{x}}{x}$

Step 1.4.2.3

Cancel the common factor.

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{x (- 3 x)}{x \cdot 1} + \frac{\frac{\sin (x)}{x}}{x}$

Step 1.4.2.4

Rewrite the expression.

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{- 3 x}{1} + \frac{\frac{\sin (x)}{x}}{x}$

Step 1.4.2.5

Divide $- 3 x$ by $1$ .

$\frac{d y}{d x} + \frac{3 y}{x} = - 3 x + \frac{\frac{\sin (x)}{x}}{x}$

Step 1.5

Factor $y$ out of $\frac{3 y}{x}$ .

$\frac{d y}{d x} + y \frac{3}{x} = - 3 x + \frac{\frac{\sin (x)}{x}}{x}$

Step 1.6

Reorder $y$ and $\frac{3}{x}$ .

$\frac{d y}{d x} + \frac{3}{x} y = - 3 x + \frac{\frac{\sin (x)}{x}}{x}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{3}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{3}{x} d x}$

Step 2.2

Integrate $\frac{3}{x}$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{3 \int \frac{1}{x} d x}$

Step 2.2.2

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

$e^{3 (\ln (| x |) + C)}$

Step 2.2.3

Simplify.

$e^{3 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{3 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{3})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{3}$

Step 3

Multiply each term by the integrating factor $x^{3}$ .

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Step 3.1

Multiply each term by $x^{3}$ .

$x^{3} \frac{d y}{d x} + x^{3} (\frac{3}{x} y) = x^{3} (- 3 x) + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{3}{x}$ and $y$ .

$x^{3} \frac{d y}{d x} + x^{3} \frac{3 y}{x} = x^{3} (- 3 x) + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

Factor $x$ out of $x^{3}$ .

$x^{3} \frac{d y}{d x} + x \cdot x^{2} \frac{3 y}{x} = x^{3} (- 3 x) + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.2.2.2

Cancel the common factor.

$x^{3} \frac{d y}{d x} + x \cdot x^{2} \frac{3 y}{x} = x^{3} (- 3 x) + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.2.2.3

Rewrite the expression.

$x^{3} \frac{d y}{d x} + x^{2} (3 y) = x^{3} (- 3 x) + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$x^{3} \frac{d y}{d x} + 3 x^{2} y = x^{3} (- 3 x) + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite using the commutative property of multiplication.

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{3} x + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.3.2

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 3.3.2.1

Move $x$ .

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 (x \cdot x^{3}) + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.3.2.2

Multiply $x$ by $x^{3}$ .

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Step 3.3.2.2.1

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

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 (x^{1} x^{3}) + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.3.2.2.2

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

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{1 + 3} + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.3.2.3

Add $1$ and $3$ .

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + x^{3} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.3.3

Cancel the common factor of $x$ .

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Step 3.3.3.1

Factor $x$ out of $x^{3}$ .

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + x \cdot x^{2} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.3.3.2

Cancel the common factor.

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + x \cdot x^{2} \frac{\frac{\sin (x)}{x}}{x}$

Step 3.3.3.3

Rewrite the expression.

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + x^{2} \frac{\sin (x)}{x}$

Step 3.3.4

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

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + \frac{x^{2} \sin (x)}{x}$

Step 3.3.5

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

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Step 3.3.5.1

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

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + \frac{x (x \sin (x))}{x}$

Step 3.3.5.2

Cancel the common factors.

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Step 3.3.5.2.1

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

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + \frac{x (x \sin (x))}{x^{1}}$

Step 3.3.5.2.2

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

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + \frac{x (x \sin (x))}{x \cdot 1}$

Step 3.3.5.2.3

Cancel the common factor.

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + \frac{x (x \sin (x))}{x \cdot 1}$

Step 3.3.5.2.4

Rewrite the expression.

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + \frac{x \sin (x)}{1}$

Step 3.3.5.2.5

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

$x^{3} \frac{d y}{d x} + 3 x^{2} y = - 3 x^{4} + x \sin (x)$

Step 4

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

$\frac{d}{d x} [x^{3} y] = - 3 x^{4} + x \sin (x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{3} y] d x = \int - 3 x^{4} + x \sin (x) d x$

Step 6

Integrate the left side.

$x^{3} y = \int - 3 x^{4} + x \sin (x) d x$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$x^{3} y = \int - 3 x^{4} d x + \int x \sin (x) d x$

Step 7.2

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$x^{3} y = - 3 \int x^{4} d x + \int x \sin (x) d x$

Step 7.3

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$x^{3} y = - 3 (\frac{1}{5} x^{5} + C) + \int x \sin (x) d x$

Step 7.4

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

$x^{3} y = - 3 (\frac{1}{5} x^{5} + C) + x (- \cos (x)) - \int - \cos (x) d x$

Step 7.5

Combine $\frac{1}{5}$ and $x^{5}$ .

$x^{3} y = - 3 (\frac{x^{5}}{5} + C) + x (- \cos (x)) - \int - \cos (x) d x$

Step 7.6

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

$x^{3} y = - 3 (\frac{x^{5}}{5} + C) + x (- \cos (x)) - - \int \cos (x) d x$

Step 7.7

Simplify.

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Step 7.7.1

Multiply $- 1$ by $- 1$ .

$x^{3} y = - 3 (\frac{x^{5}}{5} + C) + x (- \cos (x)) + 1 \int \cos (x) d x$

Step 7.7.2

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

$x^{3} y = - 3 (\frac{x^{5}}{5} + C) + x (- \cos (x)) + \int \cos (x) d x$

Step 7.8

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

$x^{3} y = - 3 (\frac{x^{5}}{5} + C) + x (- \cos (x)) + \sin (x) + C$

Step 7.9

Simplify.

$x^{3} y = - \frac{3 x^{5}}{5} - x \cos (x) + \sin (x) + C$

Step 7.10

Reorder terms.

$x^{3} y = - \frac{3}{5} x^{5} - x \cos (x) + \sin (x) + C$

Step 8

Divide each term in $x^{3} y = - \frac{3}{5} x^{5} - x \cos (x) + \sin (x) + C$ by $x^{3}$ and simplify.

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Step 8.1

Divide each term in $x^{3} y = - \frac{3}{5} x^{5} - x \cos (x) + \sin (x) + C$ by $x^{3}$ .

$\frac{x^{3} y}{x^{3}} = \frac{- \frac{3}{5} x^{5}}{x^{3}} + \frac{- x \cos (x)}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $x^{3}$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{x^{3} y}{x^{3}} = \frac{- \frac{3}{5} x^{5}}{x^{3}} + \frac{- x \cos (x)}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{3}{5} x^{5}}{x^{3}} + \frac{- x \cos (x)}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

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

Cancel the common factor of $x^{5}$ and $x^{3}$ .

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Step 8.3.1.1.1

Factor $x^{3}$ out of $- \frac{3}{5} x^{5}$ .

$y = \frac{x^{3} (- \frac{3}{5} x^{2})}{x^{3}} + \frac{- x \cos (x)}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

$y = \frac{x^{3} (- \frac{3}{5} x^{2})}{x^{3} \cdot 1} + \frac{- x \cos (x)}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{3} (- \frac{3}{5} x^{2})}{x^{3} \cdot 1} + \frac{- x \cos (x)}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.1.2.3

Rewrite the expression.

$y = \frac{- \frac{3}{5} x^{2}}{1} + \frac{- x \cos (x)}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.1.2.4

Divide $- \frac{3}{5} x^{2}$ by $1$ .

$y = - \frac{3}{5} x^{2} + \frac{- x \cos (x)}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.2

Combine $x^{2}$ and $\frac{3}{5}$ .

$y = - \frac{x^{2} \cdot 3}{5} + \frac{- x \cos (x)}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.3

Move $3$ to the left of $x^{2}$ .

$y = - \frac{3 \cdot x^{2}}{5} + \frac{- x \cos (x)}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.4

Cancel the common factor of $x$ and $x^{3}$ .

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Step 8.3.1.4.1

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

$y = - \frac{3 x^{2}}{5} + \frac{x (- \cos (x))}{x^{3}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.4.2

Cancel the common factors.

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Step 8.3.1.4.2.1

Factor $x$ out of $x^{3}$ .

$y = - \frac{3 x^{2}}{5} + \frac{x (- \cos (x))}{x \cdot x^{2}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.4.2.2

Cancel the common factor.

$y = - \frac{3 x^{2}}{5} + \frac{x (- \cos (x))}{x \cdot x^{2}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.4.2.3

Rewrite the expression.

$y = - \frac{3 x^{2}}{5} + \frac{- \cos (x)}{x^{2}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.5

Move the negative in front of the fraction.

$y = - \frac{3 x^{2}}{5} - \frac{\cos (x)}{x^{2}} + \frac{\sin (x)}{x^{3}} + \frac{C}{x^{3}}$