Calculus Examples

$x \frac{d y}{d x} - 2 y = 2 x^{4}$ , $y (2) = 8$

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 $x \frac{d y}{d x} - 2 y = 2 x^{4}$ by $x$ .

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

Factor $x$ out of $2 x^{4}$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Cancel the common factor.

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

Step 1.3.2.4

Rewrite the expression.

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

Step 1.3.2.5

Divide $2 x^{3}$ by $1$ .

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

Step 1.4

Factor $y$ out of $\frac{- 2 y}{x}$ .

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

Step 1.5

Reorder $y$ and $\frac{- 2}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 2}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{- 2}{x} d x}$

Step 2.2

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

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Step 2.2.1

Split the fraction into multiple fractions.

$e^{\int - \frac{2}{x} d x}$

Step 2.2.2

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

$e^{- \int \frac{2}{x} d x}$

Step 2.2.3

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

$e^{- (2 \int \frac{1}{x} d x)}$

Step 2.2.4

Multiply $2$ by $- 1$ .

$e^{- 2 \int \frac{1}{x} d x}$

Step 2.2.5

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

$e^{- 2 (\ln (| x |) + C)}$

Step 2.2.6

Simplify.

$e^{- 2 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- 2 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 2})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 2}$

Step 2.6

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

$\frac{1}{x^{2}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{2}}$ .

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Step 3.1

Multiply each term by $\frac{1}{x^{2}}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

Step 3.2.5

Multiply $- \frac{1}{x^{2}} \cdot \frac{2 y}{x}$ .

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Step 3.2.5.1

Multiply $\frac{2 y}{x}$ by $\frac{1}{x^{2}}$ .

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

Step 3.2.5.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.2.5.2.1

Multiply $x$ by $x^{2}$ .

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Step 3.2.5.2.1.1

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

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

Step 3.2.5.2.1.2

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

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

Step 3.2.5.2.2

Add $1$ and $2$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

Cancel the common factor.

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

Step 3.5.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify the answer.

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Step 7.3.1

Rewrite $2 (\frac{1}{2} x^{2} + C)$ as $2 (\frac{1}{2}) x^{2} + C$ .

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

Step 7.3.2

Simplify.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Cancel the common factor of $2$ .

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Step 7.3.2.2.1

Cancel the common factor.

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

Step 7.3.2.2.2

Rewrite the expression.

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

Step 7.3.2.3

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

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

Step 8

Solve for $y$ .

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Step 8.1

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

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

Step 8.2

Multiply both sides by $x^{2}$ .

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

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

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

Rewrite the expression.

$y = (x^{2} + C) x^{2}$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $(x^{2} + C) x^{2}$ .

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Step 8.3.2.1.1

Apply the distributive property.

$y = x^{2} x^{2} + C x^{2}$

Step 8.3.2.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 8.3.2.1.2.1

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

$y = x^{2 + 2} + C x^{2}$

Step 8.3.2.1.2.2

Add $2$ and $2$ .

$y = x^{4} + C x^{2}$

Step 9

Use the initial condition to find the value of $C$ by substituting $2$ for $x$ and $8$ for $y$ in $y = x^{4} + C x^{2}$ .

$8 = 2^{4} + C \cdot 2^{2}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $2^{4} + C \cdot 2^{2} = 8$ .

$2^{4} + C \cdot 2^{2} = 8$

Step 10.2

Simplify each term.

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Step 10.2.1

Raise $2$ to the power of $4$ .

$16 + C \cdot 2^{2} = 8$

Step 10.2.2

Raise $2$ to the power of $2$ .

$16 + C \cdot 4 = 8$

Step 10.2.3

Move $4$ to the left of $C$ .

$16 + 4 C = 8$

Step 10.3

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

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Step 10.3.1

Subtract $16$ from both sides of the equation.

$4 C = 8 - 16$

Step 10.3.2

Subtract $16$ from $8$ .

$4 C = - 8$

Step 10.4

Divide each term in $4 C = - 8$ by $4$ and simplify.

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Step 10.4.1

Divide each term in $4 C = - 8$ by $4$ .

$\frac{4 C}{4} = \frac{- 8}{4}$

Step 10.4.2

Simplify the left side.

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Step 10.4.2.1

Cancel the common factor of $4$ .

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Step 10.4.2.1.1

Cancel the common factor.

$\frac{4 C}{4} = \frac{- 8}{4}$

Step 10.4.2.1.2

Divide $C$ by $1$ .

$C = \frac{- 8}{4}$

Step 10.4.3

Simplify the right side.

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Step 10.4.3.1

Divide $- 8$ by $4$ .

$C = - 2$

Step 11

Substitute $- 2$ for $C$ in $y = x^{4} + C x^{2}$ and simplify.

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Step 11.1

Substitute $- 2$ for $C$ .

$y = x^{4} - 2 x^{2}$