Calculus Examples

$x \frac{d y}{d x} - 6 y = x^{2}$ , $y (1) = 10$

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 $x \frac{d y}{d x} - 6 y = x^{2}$ by $x$ .

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

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

$\frac{d y}{d x} + \frac{- 6 y}{x} = \frac{x \cdot x}{x}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Cancel the common factor.

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

Step 1.3.2.4

Rewrite the expression.

$\frac{d y}{d x} + \frac{- 6 y}{x} = \frac{x}{1}$

Step 1.3.2.5

Divide $x$ by $1$ .

$\frac{d y}{d x} + \frac{- 6 y}{x} = x$

Step 1.4

Factor $y$ out of $\frac{- 6 y}{x}$ .

$\frac{d y}{d x} + y \frac{- 6}{x} = x$

Step 1.5

Reorder $y$ and $\frac{- 6}{x}$ .

$\frac{d y}{d x} + \frac{- 6}{x} y = x$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 6}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{- 6}{x} d x}$

Step 2.2

Integrate $\frac{- 6}{x}$ .

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Step 2.2.1

Split the fraction into multiple fractions.

$e^{\int - \frac{6}{x} d x}$

Step 2.2.2

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

$e^{- \int \frac{6}{x} d x}$

Step 2.2.3

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$e^{- (6 \int \frac{1}{x} d x)}$

Step 2.2.4

Multiply $6$ by $- 1$ .

$e^{- 6 \int \frac{1}{x} d x}$

Step 2.2.5

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

$e^{- 6 (\ln (| x |) + C)}$

Step 2.2.6

Simplify.

$e^{- 6 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- 6 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 6})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 6}$

Step 2.6

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

$\frac{1}{x^{6}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{6}}$ .

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Step 3.1

Multiply each term by $\frac{1}{x^{6}}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{1}{x^{6}}$ and $\frac{d y}{d x}$ .

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{x^{6}} - \frac{1}{x^{6}} (\frac{6}{x} y) = \frac{1}{x^{6}} x$

Step 3.2.4

Combine $\frac{6}{x}$ and $y$ .

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

Step 3.2.5

Multiply $- \frac{1}{x^{6}} \cdot \frac{6 y}{x}$ .

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Step 3.2.5.1

Multiply $\frac{6 y}{x}$ by $\frac{1}{x^{6}}$ .

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

Step 3.2.5.2

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 3.2.5.2.1

Multiply $x$ by $x^{6}$ .

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Step 3.2.5.2.1.1

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

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

Step 3.2.5.2.1.2

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

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

Step 3.2.5.2.2

Add $1$ and $6$ .

$\frac{\frac{d y}{d x}}{x^{6}} - \frac{6 y}{x^{7}} = \frac{1}{x^{6}} x$

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Factor $x$ out of $x^{6}$ .

$\frac{\frac{d y}{d x}}{x^{6}} - \frac{6 y}{x^{7}} = \frac{1}{x \cdot x^{5}} x$

Step 3.3.2

Cancel the common factor.

$\frac{\frac{d y}{d x}}{x^{6}} - \frac{6 y}{x^{7}} = \frac{1}{x \cdot x^{5}} x$

Step 3.3.3

Rewrite the expression.

$\frac{\frac{d y}{d x}}{x^{6}} - \frac{6 y}{x^{7}} = \frac{1}{x^{5}}$

Step 4

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

$\frac{d}{d x} [\frac{1}{x^{6}} y] = \frac{1}{x^{5}}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{6}} y] d x = \int \frac{1}{x^{5}} d x$

Step 6

Integrate the left side.

$\frac{1}{x^{6}} y = \int \frac{1}{x^{5}} d x$

Step 7

Integrate the right side.

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Step 7.1

Apply basic rules of exponents.

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Step 7.1.1

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

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

Step 7.1.2

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

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Step 7.1.2.1

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

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

Step 7.1.2.2

Multiply $5$ by $- 1$ .

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

Step 7.2

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

$\frac{1}{x^{6}} y = - \frac{1}{4} x^{- 4} + C$

Step 7.3

Simplify the answer.

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Step 7.3.1

Rewrite $- \frac{1}{4} x^{- 4} + C$ as $- \frac{1}{4} \cdot \frac{1}{x^{4}} + C$ .

$\frac{1}{x^{6}} y = - \frac{1}{4} \cdot \frac{1}{x^{4}} + C$

Step 7.3.2

Simplify.

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Step 7.3.2.1

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

$\frac{1}{x^{6}} y = - \frac{1}{x^{4} \cdot 4} + C$

Step 7.3.2.2

Move $4$ to the left of $x^{4}$ .

$\frac{1}{x^{6}} y = - \frac{1}{4 x^{4}} + C$

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

$\frac{1}{x^{6}} y + \frac{1}{4 x^{4}} = C$

Step 8.1.2

Subtract $C$ from both sides of the equation.

$\frac{1}{x^{6}} y + \frac{1}{4 x^{4}} - C = 0$

Step 8.1.3

Combine $\frac{1}{x^{6}}$ and $y$ .

$\frac{y}{x^{6}} + \frac{1}{4 x^{4}} - C = 0$

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

$\frac{y}{x^{6}} - C = - \frac{1}{4 x^{4}}$

Step 8.2.2

Add $C$ to both sides of the equation.

$\frac{y}{x^{6}} = - \frac{1}{4 x^{4}} + C$

Step 8.3

Multiply both sides by $x^{6}$ .

$\frac{y}{x^{6}} x^{6} = (- \frac{1}{4 x^{4}} + C) x^{6}$

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 $x^{6}$ .

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Step 8.4.1.1.1

Cancel the common factor.

$\frac{y}{x^{6}} x^{6} = (- \frac{1}{4 x^{4}} + C) x^{6}$

Step 8.4.1.1.2

Rewrite the expression.

$y = (- \frac{1}{4 x^{4}} + C) x^{6}$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(- \frac{1}{4 x^{4}} + C) x^{6}$ .

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Step 8.4.2.1.1

Apply the distributive property.

$y = - \frac{1}{4 x^{4}} x^{6} + C x^{6}$

Step 8.4.2.1.2

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

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Step 8.4.2.1.2.1

Move the leading negative in $- \frac{1}{4 x^{4}}$ into the numerator.

$y = \frac{- 1}{4 x^{4}} x^{6} + C x^{6}$

Step 8.4.2.1.2.2

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

$y = \frac{- 1}{x^{4} \cdot 4} x^{6} + C x^{6}$

Step 8.4.2.1.2.3

Factor $x^{4}$ out of $x^{6}$ .

$y = \frac{- 1}{x^{4} \cdot 4} (x^{4} x^{2}) + C x^{6}$

Step 8.4.2.1.2.4

Cancel the common factor.

$y = \frac{- 1}{x^{4} \cdot 4} (x^{4} x^{2}) + C x^{6}$

Step 8.4.2.1.2.5

Rewrite the expression.

$y = \frac{- 1}{4} x^{2} + C x^{6}$

Step 8.4.2.1.3

Combine $\frac{- 1}{4}$ and $x^{2}$ .

$y = \frac{- x^{2}}{4} + C x^{6}$

Step 8.4.2.1.4

Simplify the expression.

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Step 8.4.2.1.4.1

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{4} + C x^{6}$

Step 8.4.2.1.4.2

Reorder $- \frac{x^{2}}{4}$ and $C x^{6}$ .

$y = C x^{6} - \frac{x^{2}}{4}$

Step 9

Use the initial condition to find the value of $C$ by substituting $1$ for $x$ and $10$ for $y$ in $y = C x^{6} - \frac{x^{2}}{4}$ .

$10 = C \cdot 1^{6} - \frac{1^{2}}{4}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $C \cdot 1^{6} - \frac{1^{2}}{4} = 10$ .

$C \cdot 1^{6} - \frac{1^{2}}{4} = 10$

Step 10.2

Simplify each term.

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Step 10.2.1

One to any power is one.

$C \cdot 1 - \frac{1^{2}}{4} = 10$

Step 10.2.2

Multiply $C$ by $1$ .

$C - \frac{1^{2}}{4} = 10$

Step 10.2.3

One to any power is one.

$C - \frac{1}{4} = 10$

Step 10.3

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

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Step 10.3.1

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

$C = 10 + \frac{1}{4}$

Step 10.3.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$C = 10 \cdot \frac{4}{4} + \frac{1}{4}$

Step 10.3.3

Combine $10$ and $\frac{4}{4}$ .

$C = \frac{10 \cdot 4}{4} + \frac{1}{4}$

Step 10.3.4

Combine the numerators over the common denominator.

$C = \frac{10 \cdot 4 + 1}{4}$

Step 10.3.5

Simplify the numerator.

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Step 10.3.5.1

Multiply $10$ by $4$ .

$C = \frac{40 + 1}{4}$

Step 10.3.5.2

Add $40$ and $1$ .

$C = \frac{41}{4}$

Step 11

Substitute $\frac{41}{4}$ for $C$ in $y = C x^{6} - \frac{x^{2}}{4}$ and simplify.

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Step 11.1

Substitute $\frac{41}{4}$ for $C$ .

$y = \frac{41}{4} x^{6} - \frac{x^{2}}{4}$

Step 11.2

Combine $\frac{41}{4}$ and $x^{6}$ .

$y = \frac{41 x^{6}}{4} - \frac{x^{2}}{4}$