Calculus Examples

$x \frac{d y}{d x} - 4 y + 2 x = 0$ , $y (1) = 10$

Step 1

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

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Step 1.1

Subtract $2 x$ from both sides of the equation.

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

Step 1.2

Divide each term in $x \frac{d y}{d x} - 4 y = - 2 x$ by $x$ .

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

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

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

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Divide $- 2$ by $1$ .

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

Step 1.5

Factor $y$ out of $\frac{- 4 y}{x}$ .

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

Step 1.6

Reorder $y$ and $\frac{- 4}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 4}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{- 4}{x} d x}$

Step 2.2

Integrate $\frac{- 4}{x}$ .

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Step 2.2.1

Split the fraction into multiple fractions.

$e^{\int - \frac{4}{x} d x}$

Step 2.2.2

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

$e^{- \int \frac{4}{x} d x}$

Step 2.2.3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$e^{- (4 \int \frac{1}{x} d x)}$

Step 2.2.4

Multiply $4$ by $- 1$ .

$e^{- 4 \int \frac{1}{x} d x}$

Step 2.2.5

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

$e^{- 4 (\ln (| x |) + C)}$

Step 2.2.6

Simplify.

$e^{- 4 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- 4 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 4})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 4}$

Step 2.6

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

$\frac{1}{x^{4}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{4}}$ .

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

$\frac{\frac{d y}{d x}}{x^{4}} - \frac{1}{x^{4}} \cdot \frac{4 y}{x} = \frac{1}{x^{4}} \cdot - 2$

Step 3.2.5

Multiply $- \frac{1}{x^{4}} \cdot \frac{4 y}{x}$ .

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Step 3.2.5.1

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

$\frac{\frac{d y}{d x}}{x^{4}} - \frac{4 y}{x \cdot x^{4}} = \frac{1}{x^{4}} \cdot - 2$

Step 3.2.5.2

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 3.2.5.2.1

Multiply $x$ by $x^{4}$ .

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Step 3.2.5.2.1.1

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

$\frac{\frac{d y}{d x}}{x^{4}} - \frac{4 y}{x^{1} x^{4}} = \frac{1}{x^{4}} \cdot - 2$

Step 3.2.5.2.1.2

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

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

Step 3.2.5.2.2

Add $1$ and $4$ .

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

Step 3.3

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

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

Step 3.4

Move the negative in front of the fraction.

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

Step 4

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

$\frac{d}{d x} [\frac{1}{x^{4}} y] = - \frac{2}{x^{4}}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{4}} y] d x = \int - \frac{2}{x^{4}} d x$

Step 6

Integrate the left side.

$\frac{1}{x^{4}} y = \int - \frac{2}{x^{4}} d x$

Step 7

Integrate the right side.

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Step 7.1

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

$\frac{1}{x^{4}} y = - \int \frac{2}{x^{4}} d x$

Step 7.2

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

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

Step 7.3

Simplify the expression.

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Step 7.3.1

Multiply $2$ by $- 1$ .

$\frac{1}{x^{4}} y = - 2 \int \frac{1}{x^{4}} d x$

Step 7.3.2

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

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

Step 7.3.3

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 7.3.3.1

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

$\frac{1}{x^{4}} y = - 2 \int x^{4 \cdot - 1} d x$

Step 7.3.3.2

Multiply $4$ by $- 1$ .

$\frac{1}{x^{4}} y = - 2 \int x^{- 4} d x$

Step 7.4

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

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

Step 7.5

Simplify the answer.

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Step 7.5.1

Simplify.

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Step 7.5.1.1

Combine $x^{- 3}$ and $\frac{1}{3}$ .

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

Step 7.5.1.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 7.5.2

Simplify.

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

Step 7.5.3

Simplify.

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Step 7.5.3.1

Multiply $- 1$ by $- 2$ .

$\frac{1}{x^{4}} y = 2 \frac{1}{3 x^{3}} + C$

Step 7.5.3.2

Combine $2$ and $\frac{1}{3 x^{3}}$ .

$\frac{1}{x^{4}} y = \frac{2}{3 x^{3}} + C$

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

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

$\frac{1}{x^{4}} y - \frac{2}{3 x^{3}} = C$

Step 8.1.2

Subtract $C$ from both sides of the equation.

$\frac{1}{x^{4}} y - \frac{2}{3 x^{3}} - C = 0$

Step 8.1.3

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

$\frac{y}{x^{4}} - \frac{2}{3 x^{3}} - C = 0$

Step 8.2

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

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Step 8.2.1

Add $\frac{2}{3 x^{3}}$ to both sides of the equation.

$\frac{y}{x^{4}} - C = \frac{2}{3 x^{3}}$

Step 8.2.2

Add $C$ to both sides of the equation.

$\frac{y}{x^{4}} = \frac{2}{3 x^{3}} + C$

Step 8.3

Multiply both sides by $x^{4}$ .

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

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 $x^{4}$ .

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Step 8.4.1.1.1

Cancel the common factor.

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

Step 8.4.1.1.2

Rewrite the expression.

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

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(\frac{2}{3 x^{3}} + C) x^{4}$ .

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Step 8.4.2.1.1

Apply the distributive property.

$y = \frac{2}{3 x^{3}} x^{4} + C x^{4}$

Step 8.4.2.1.2

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

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Step 8.4.2.1.2.1

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

$y = \frac{2}{x^{3} \cdot 3} x^{4} + C x^{4}$

Step 8.4.2.1.2.2

Factor $x^{3}$ out of $x^{4}$ .

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

Step 8.4.2.1.2.3

Cancel the common factor.

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

Step 8.4.2.1.2.4

Rewrite the expression.

$y = \frac{2}{3} x + C x^{4}$

Step 8.4.2.1.3

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

$y = \frac{2 x}{3} + C x^{4}$

Step 8.4.2.1.4

Reorder $\frac{2 x}{3}$ and $C x^{4}$ .

$y = C x^{4} + \frac{2 x}{3}$

Step 9

Use the initial condition to find the value of $C$ by substituting $1$ for $x$ and $10$ for $y$ in $y = C x^{4} + \frac{2 x}{3}$ .

$10 = C \cdot 1^{4} + \frac{2 \cdot 1}{3}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $C \cdot 1^{4} + \frac{2 \cdot 1}{3} = 10$ .

$C \cdot 1^{4} + \frac{2 \cdot 1}{3} = 10$

Step 10.2

Simplify $C \cdot 1^{4} + \frac{2 \cdot 1}{3}$ .

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Step 10.2.1

Multiply $2$ by $1$ .

$C \cdot 1^{4} + \frac{2}{3} = 10$

Step 10.2.2

Simplify each term.

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Step 10.2.2.1

One to any power is one.

$C \cdot 1 + \frac{2}{3} = 10$

Step 10.2.2.2

Multiply $C$ by $1$ .

$C + \frac{2}{3} = 10$

Step 10.3

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

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Step 10.3.1

Subtract $\frac{2}{3}$ from both sides of the equation.

$C = 10 - \frac{2}{3}$

Step 10.3.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$C = 10 \cdot \frac{3}{3} - \frac{2}{3}$

Step 10.3.3

Combine $10$ and $\frac{3}{3}$ .

$C = \frac{10 \cdot 3}{3} - \frac{2}{3}$

Step 10.3.4

Combine the numerators over the common denominator.

$C = \frac{10 \cdot 3 - 2}{3}$

Step 10.3.5

Simplify the numerator.

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Step 10.3.5.1

Multiply $10$ by $3$ .

$C = \frac{30 - 2}{3}$

Step 10.3.5.2

Subtract $2$ from $30$ .

$C = \frac{28}{3}$

Step 11

Substitute $\frac{28}{3}$ for $C$ in $y = C x^{4} + \frac{2 x}{3}$ and simplify.

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Step 11.1

Substitute $\frac{28}{3}$ for $C$ .

$y = \frac{28}{3} x^{4} + \frac{2 x}{3}$

Step 11.2

Combine $\frac{28}{3}$ and $x^{4}$ .

$y = \frac{28 x^{4}}{3} + \frac{2 x}{3}$