Calculus Examples

$x \frac{d y}{d x} = 5 y + 2 x^{3} - x$

Step 1

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

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Step 1.1

Subtract $5 y$ from both sides of the equation.

$x \frac{d y}{d x} - 5 y = 2 x^{3} - x$

Step 1.2

Divide each term in $x \frac{d y}{d x} - 5 y = 2 x^{3} - x$ by $x$ .

$\frac{x \frac{d y}{d x}}{x} + \frac{- 5 y}{x} = \frac{2 x^{3}}{x} + \frac{- x}{x}$

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} + \frac{- 5 y}{x} = \frac{2 x^{3}}{x} + \frac{- x}{x}$

Step 1.3.2

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

$\frac{d y}{d x} + \frac{- 5 y}{x} = \frac{2 x^{3}}{x} + \frac{- x}{x}$

Step 1.4

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

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Step 1.4.1

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

$\frac{d y}{d x} + \frac{- 5 y}{x} = \frac{x (2 x^{2})}{x} + \frac{- x}{x}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

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

$\frac{d y}{d x} + \frac{- 5 y}{x} = \frac{x (2 x^{2})}{x^{1}} + \frac{- x}{x}$

Step 1.4.2.2

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

$\frac{d y}{d x} + \frac{- 5 y}{x} = \frac{x (2 x^{2})}{x \cdot 1} + \frac{- x}{x}$

Step 1.4.2.3

Cancel the common factor.

$\frac{d y}{d x} + \frac{- 5 y}{x} = \frac{x (2 x^{2})}{x \cdot 1} + \frac{- x}{x}$

Step 1.4.2.4

Rewrite the expression.

$\frac{d y}{d x} + \frac{- 5 y}{x} = \frac{2 x^{2}}{1} + \frac{- x}{x}$

Step 1.4.2.5

Divide $2 x^{2}$ by $1$ .

$\frac{d y}{d x} + \frac{- 5 y}{x} = 2 x^{2} + \frac{- x}{x}$

Step 1.5

Cancel the common factor of $x$ .

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Step 1.5.1

Cancel the common factor.

$\frac{d y}{d x} + \frac{- 5 y}{x} = 2 x^{2} + \frac{- x}{x}$

Step 1.5.2

Divide $- 1$ by $1$ .

$\frac{d y}{d x} + \frac{- 5 y}{x} = 2 x^{2} - 1$

Step 1.6

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

$\frac{d y}{d x} + y \frac{- 5}{x} = 2 x^{2} - 1$

Step 1.7

Reorder $y$ and $\frac{- 5}{x}$ .

$\frac{d y}{d x} + \frac{- 5}{x} y = 2 x^{2} - 1$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 5}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{- 5}{x} d x}$

Step 2.2

Integrate $\frac{- 5}{x}$ .

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Step 2.2.1

Split the fraction into multiple fractions.

$e^{\int - \frac{5}{x} d x}$

Step 2.2.2

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

$e^{- \int \frac{5}{x} d x}$

Step 2.2.3

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$e^{- (5 \int \frac{1}{x} d x)}$

Step 2.2.4

Multiply $5$ by $- 1$ .

$e^{- 5 \int \frac{1}{x} d x}$

Step 2.2.5

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

$e^{- 5 (\ln (| x |) + C)}$

Step 2.2.6

Simplify.

$e^{- 5 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- 5 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 5})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 5}$

Step 2.6

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

$\frac{1}{x^{5}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{5}}$ .

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Step 3.1

Multiply each term by $\frac{1}{x^{5}}$ .

$\frac{1}{x^{5}} \frac{d y}{d x} + \frac{1}{x^{5}} (\frac{- 5}{x} y) = \frac{1}{x^{5}} (2 x^{2}) + \frac{1}{x^{5}} \cdot - 1$

Step 3.2

Simplify each term.

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Step 3.2.1

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

$\frac{\frac{d y}{d x}}{x^{5}} + \frac{1}{x^{5}} (\frac{- 5}{x} y) = \frac{1}{x^{5}} (2 x^{2}) + \frac{1}{x^{5}} \cdot - 1$

Step 3.2.2

Move the negative in front of the fraction.

$\frac{\frac{d y}{d x}}{x^{5}} + \frac{1}{x^{5}} (- \frac{5}{x} y) = \frac{1}{x^{5}} (2 x^{2}) + \frac{1}{x^{5}} \cdot - 1$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{1}{x^{5}} (\frac{5}{x} y) = \frac{1}{x^{5}} (2 x^{2}) + \frac{1}{x^{5}} \cdot - 1$

Step 3.2.4

Combine $\frac{5}{x}$ and $y$ .

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{1}{x^{5}} \cdot \frac{5 y}{x} = \frac{1}{x^{5}} (2 x^{2}) + \frac{1}{x^{5}} \cdot - 1$

Step 3.2.5

Multiply $- \frac{1}{x^{5}} \cdot \frac{5 y}{x}$ .

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Step 3.2.5.1

Multiply $\frac{5 y}{x}$ by $\frac{1}{x^{5}}$ .

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x \cdot x^{5}} = \frac{1}{x^{5}} (2 x^{2}) + \frac{1}{x^{5}} \cdot - 1$

Step 3.2.5.2

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 3.2.5.2.1

Multiply $x$ by $x^{5}$ .

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Step 3.2.5.2.1.1

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

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x^{1} x^{5}} = \frac{1}{x^{5}} (2 x^{2}) + \frac{1}{x^{5}} \cdot - 1$

Step 3.2.5.2.1.2

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

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x^{1 + 5}} = \frac{1}{x^{5}} (2 x^{2}) + \frac{1}{x^{5}} \cdot - 1$

Step 3.2.5.2.2

Add $1$ and $5$ .

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x^{6}} = \frac{1}{x^{5}} (2 x^{2}) + \frac{1}{x^{5}} \cdot - 1$

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x^{6}} = 2 \frac{1}{x^{5}} x^{2} + \frac{1}{x^{5}} \cdot - 1$

Step 3.3.2

Combine $2$ and $\frac{1}{x^{5}}$ .

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x^{6}} = \frac{2}{x^{5}} x^{2} + \frac{1}{x^{5}} \cdot - 1$

Step 3.3.3

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

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Step 3.3.3.1

Factor $x^{2}$ out of $x^{5}$ .

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x^{6}} = \frac{2}{x^{2} x^{3}} x^{2} + \frac{1}{x^{5}} \cdot - 1$

Step 3.3.3.2

Cancel the common factor.

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x^{6}} = \frac{2}{x^{2} x^{3}} x^{2} + \frac{1}{x^{5}} \cdot - 1$

Step 3.3.3.3

Rewrite the expression.

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x^{6}} = \frac{2}{x^{3}} + \frac{1}{x^{5}} \cdot - 1$

Step 3.3.4

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

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x^{6}} = \frac{2}{x^{3}} + \frac{- 1}{x^{5}}$

Step 3.3.5

Move the negative in front of the fraction.

$\frac{\frac{d y}{d x}}{x^{5}} - \frac{5 y}{x^{6}} = \frac{2}{x^{3}} - \frac{1}{x^{5}}$

Step 4

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

$\frac{d}{d x} [\frac{1}{x^{5}} y] = \frac{2}{x^{3}} - \frac{1}{x^{5}}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{5}} y] d x = \int \frac{2}{x^{3}} - \frac{1}{x^{5}} d x$

Step 6

Integrate the left side.

$\frac{1}{x^{5}} y = \int \frac{2}{x^{3}} - \frac{1}{x^{5}} d x$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$\frac{1}{x^{5}} y = \int \frac{2}{x^{3}} d x + \int - \frac{1}{x^{5}} d x$

Step 7.2

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

$\frac{1}{x^{5}} y = 2 \int \frac{1}{x^{3}} d x + \int - \frac{1}{x^{5}} d x$

Step 7.3

Apply basic rules of exponents.

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Step 7.3.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{x^{5}} y = 2 \int {(x^{3})}^{- 1} d x + \int - \frac{1}{x^{5}} d x$

Step 7.3.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 7.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{x^{5}} y = 2 \int x^{3 \cdot - 1} d x + \int - \frac{1}{x^{5}} d x$

Step 7.3.2.2

Multiply $3$ by $- 1$ .

$\frac{1}{x^{5}} y = 2 \int x^{- 3} d x + \int - \frac{1}{x^{5}} d x$

Step 7.4

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

$\frac{1}{x^{5}} y = 2 (- \frac{1}{2} x^{- 2} + C) + \int - \frac{1}{x^{5}} d x$

Step 7.5

Simplify.

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Step 7.5.1

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

$\frac{1}{x^{5}} y = 2 (- \frac{x^{- 2}}{2} + C) + \int - \frac{1}{x^{5}} d x$

Step 7.5.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{5}} y = 2 (- \frac{1}{2 x^{2}} + C) + \int - \frac{1}{x^{5}} d x$

Step 7.6

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

$\frac{1}{x^{5}} y = 2 (- \frac{1}{2 x^{2}} + C) - \int \frac{1}{x^{5}} d x$

Step 7.7

Apply basic rules of exponents.

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Step 7.7.1

Move $x^{5}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{x^{5}} y = 2 (- \frac{1}{2 x^{2}} + C) - \int {(x^{5})}^{- 1} d x$

Step 7.7.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 7.7.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{x^{5}} y = 2 (- \frac{1}{2 x^{2}} + C) - \int x^{5 \cdot - 1} d x$

Step 7.7.2.2

Multiply $5$ by $- 1$ .

$\frac{1}{x^{5}} y = 2 (- \frac{1}{2 x^{2}} + C) - \int x^{- 5} d x$

Step 7.8

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

$\frac{1}{x^{5}} y = 2 (- \frac{1}{2 x^{2}} + C) - (- \frac{1}{4} x^{- 4} + C)$

Step 7.9

Simplify.

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Step 7.9.1

Simplify.

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Step 7.9.1.1

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

$\frac{1}{x^{5}} y = 2 (- \frac{1}{2 x^{2}} + C) - (- \frac{x^{- 4}}{4} + C)$

Step 7.9.1.2

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{5}} y = 2 (- \frac{1}{2 x^{2}} + C) - (- \frac{1}{4 x^{4}} + C)$

Step 7.9.2

Simplify.

$\frac{1}{x^{5}} y = \frac{- 1}{x^{2}} - - \frac{1}{4 x^{4}} + C$

Step 7.9.3

Simplify.

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Step 7.9.3.1

Move the negative in front of the fraction.

$\frac{1}{x^{5}} y = - \frac{1}{x^{2}} - - \frac{1}{4 x^{4}} + C$

Step 7.9.3.2

Multiply $- 1$ by $- 1$ .

$\frac{1}{x^{5}} y = - \frac{1}{x^{2}} + 1 \frac{1}{4 x^{4}} + C$

Step 7.9.3.3

Multiply $\frac{1}{4 x^{4}}$ by $1$ .

$\frac{1}{x^{5}} y = - \frac{1}{x^{2}} + \frac{1}{4 x^{4}} + C$

Step 8

Solve for $y$ .

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Step 8.1

Move all terms containing variables to the left side of the equation.

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Step 8.1.1

Add $\frac{1}{x^{2}}$ to both sides of the equation.

$\frac{1}{x^{5}} y + \frac{1}{x^{2}} = \frac{1}{4 x^{4}} + C$

Step 8.1.2

Subtract $\frac{1}{4 x^{4}}$ from both sides of the equation.

$\frac{1}{x^{5}} y + \frac{1}{x^{2}} - \frac{1}{4 x^{4}} = C$

Step 8.1.3

Subtract $C$ from both sides of the equation.

$\frac{1}{x^{5}} y + \frac{1}{x^{2}} - \frac{1}{4 x^{4}} - C = 0$

Step 8.1.4

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

$\frac{y}{x^{5}} + \frac{1}{x^{2}} - \frac{1}{4 x^{4}} - C = 0$

Step 8.2

Move all terms not containing $y$ to the right side of the equation.

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Step 8.2.1

Subtract $\frac{1}{x^{2}}$ from both sides of the equation.

$\frac{y}{x^{5}} - \frac{1}{4 x^{4}} - C = - \frac{1}{x^{2}}$

Step 8.2.2

Add $\frac{1}{4 x^{4}}$ to both sides of the equation.

$\frac{y}{x^{5}} - C = - \frac{1}{x^{2}} + \frac{1}{4 x^{4}}$

Step 8.2.3

Add $C$ to both sides of the equation.

$\frac{y}{x^{5}} = - \frac{1}{x^{2}} + \frac{1}{4 x^{4}} + C$

Step 8.3

Multiply both sides by $x^{5}$ .

$\frac{y}{x^{5}} x^{5} = (- \frac{1}{x^{2}} + \frac{1}{4 x^{4}} + C) x^{5}$

Step 8.4

Simplify.

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

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

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Step 8.4.1.1.1

Cancel the common factor.

$\frac{y}{x^{5}} x^{5} = (- \frac{1}{x^{2}} + \frac{1}{4 x^{4}} + C) x^{5}$

Step 8.4.1.1.2

Rewrite the expression.

$y = (- \frac{1}{x^{2}} + \frac{1}{4 x^{4}} + C) x^{5}$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(- \frac{1}{x^{2}} + \frac{1}{4 x^{4}} + C) x^{5}$ .

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Step 8.4.2.1.1

Apply the distributive property.

$y = - \frac{1}{x^{2}} x^{5} + \frac{1}{4 x^{4}} x^{5} + C x^{5}$

Step 8.4.2.1.2

Simplify.

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Step 8.4.2.1.2.1

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

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Step 8.4.2.1.2.1.1

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

$y = \frac{- 1}{x^{2}} x^{5} + \frac{1}{4 x^{4}} x^{5} + C x^{5}$

Step 8.4.2.1.2.1.2

Factor $x^{2}$ out of $x^{5}$ .

$y = \frac{- 1}{x^{2}} (x^{2} x^{3}) + \frac{1}{4 x^{4}} x^{5} + C x^{5}$

Step 8.4.2.1.2.1.3

Cancel the common factor.

$y = \frac{- 1}{x^{2}} (x^{2} x^{3}) + \frac{1}{4 x^{4}} x^{5} + C x^{5}$

Step 8.4.2.1.2.1.4

Rewrite the expression.

$y = - 1 x^{3} + \frac{1}{4 x^{4}} x^{5} + C x^{5}$

Step 8.4.2.1.2.2

Cancel the common factor of $x^{4}$ .

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Step 8.4.2.1.2.2.1

Factor $x^{4}$ out of $4 x^{4}$ .

$y = - 1 x^{3} + \frac{1}{x^{4} \cdot 4} x^{5} + C x^{5}$

Step 8.4.2.1.2.2.2

Factor $x^{4}$ out of $x^{5}$ .

$y = - 1 x^{3} + \frac{1}{x^{4} \cdot 4} (x^{4} x) + C x^{5}$

Step 8.4.2.1.2.2.3

Cancel the common factor.

$y = - 1 x^{3} + \frac{1}{x^{4} \cdot 4} (x^{4} x) + C x^{5}$

Step 8.4.2.1.2.2.4

Rewrite the expression.

$y = - 1 x^{3} + \frac{1}{4} x + C x^{5}$

Step 8.4.2.1.2.3

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

$y = - 1 x^{3} + \frac{x}{4} + C x^{5}$

Step 8.4.2.1.3

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

$y = - x^{3} + \frac{x}{4} + C x^{5}$

Step 8.4.2.1.4

Move $\frac{x}{4}$ .

$y = - x^{3} + C x^{5} + \frac{x}{4}$

Step 8.4.2.1.5

Reorder $- x^{3}$ and $C x^{5}$ .

$y = C x^{5} - x^{3} + \frac{x}{4}$