Calculus Examples

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

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 $4 x \frac{d y}{d x} - y = 3 x^{2}$ by $4 x$ .

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

Step 1.2

Cancel the common factor of $4$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $x$ out of $4 x$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

$\frac{d y}{d x} + \frac{- y}{4 x} = \frac{3 x}{4}$

Step 1.5

Reorder terms.

$\frac{d y}{d x} + \frac{- y}{4 x} = \frac{3}{4} x$

Step 1.6

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

$\frac{d y}{d x} + y \frac{- 1}{4 x} = \frac{3}{4} x$

Step 1.7

Reorder $y$ and $\frac{- 1}{4 x}$ .

$\frac{d y}{d x} + \frac{- 1}{4 x} y = \frac{3}{4} x$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 1}{4 x}$ .

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $\frac{- 1}{4 x}$ .

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Step 2.2.1

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

Step 2.2.3

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

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

Step 2.2.4

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

$e^{- \frac{1}{4} (\ln (| x |) + C)}$

Step 2.2.5

Simplify.

$e^{- \frac{1}{4} \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

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

Step 2.5

Exponentiation and log are inverse functions.

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

Step 2.6

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

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

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Step 3.2.5.2.1

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

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

Step 3.2.5.2.2

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

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Step 3.2.5.2.2.1

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

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

Step 3.2.5.2.2.2

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

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

Step 3.2.5.2.3

Write $1$ as a fraction with a common denominator.

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

Step 3.2.5.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5.2.5

Add $1$ and $4$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Combine.

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

Step 3.5

Multiply $3$ by $1$ .

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

Step 3.6

Combine $\frac{3}{4 x^{\frac{1}{4}}}$ and $x$ .

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

Step 3.7

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

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

Step 3.8

Multiply $x$ by $x^{- \frac{1}{4}}$ by adding the exponents.

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Step 3.8.1

Move $x^{- \frac{1}{4}}$ .

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

Step 3.8.2

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

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Step 3.8.2.1

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

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

Step 3.8.2.2

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

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

Step 3.8.3

Write $1$ as a fraction with a common denominator.

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

Step 3.8.4

Combine the numerators over the common denominator.

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

Step 3.8.5

Add $- 1$ and $4$ .

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

Step 4

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

$\frac{d}{d x} [\frac{1}{x^{\frac{1}{4}}} y] = \frac{3 x^{\frac{3}{4}}}{4}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{\frac{1}{4}}} y] d x = \int \frac{3 x^{\frac{3}{4}}}{4} d x$

Step 6

Integrate the left side.

$\frac{1}{x^{\frac{1}{4}}} y = \int \frac{3 x^{\frac{3}{4}}}{4} d x$

Step 7

Integrate the right side.

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Step 7.1

Since $\frac{3}{4}$ is constant with respect to $x$ , move $\frac{3}{4}$ out of the integral.

$\frac{1}{x^{\frac{1}{4}}} y = \frac{3}{4} \int x^{\frac{3}{4}} d x$

Step 7.2

By the Power Rule, the integral of $x^{\frac{3}{4}}$ with respect to $x$ is $\frac{4}{7} x^{\frac{7}{4}}$ .

$\frac{1}{x^{\frac{1}{4}}} y = \frac{3}{4} (\frac{4}{7} x^{\frac{7}{4}} + C)$

Step 7.3

Simplify the answer.

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Step 7.3.1

Rewrite $\frac{3}{4} (\frac{4}{7} x^{\frac{7}{4}} + C)$ as $\frac{3}{4} \cdot \frac{4}{7} x^{\frac{7}{4}} + C$ .

$\frac{1}{x^{\frac{1}{4}}} y = \frac{3}{4} \cdot \frac{4}{7} x^{\frac{7}{4}} + C$

Step 7.3.2

Simplify.

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Step 7.3.2.1

Multiply $\frac{3}{4}$ by $\frac{4}{7}$ .

$\frac{1}{x^{\frac{1}{4}}} y = \frac{3 \cdot 4}{4 \cdot 7} x^{\frac{7}{4}} + C$

Step 7.3.2.2

Multiply $3$ by $4$ .

$\frac{1}{x^{\frac{1}{4}}} y = \frac{12}{4 \cdot 7} x^{\frac{7}{4}} + C$

Step 7.3.2.3

Multiply $4$ by $7$ .

$\frac{1}{x^{\frac{1}{4}}} y = \frac{12}{28} x^{\frac{7}{4}} + C$

Step 7.3.2.4

Cancel the common factor of $12$ and $28$ .

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Step 7.3.2.4.1

Factor $4$ out of $12$ .

$\frac{1}{x^{\frac{1}{4}}} y = \frac{4 (3)}{28} x^{\frac{7}{4}} + C$

Step 7.3.2.4.2

Cancel the common factors.

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Step 7.3.2.4.2.1

Factor $4$ out of $28$ .

$\frac{1}{x^{\frac{1}{4}}} y = \frac{4 \cdot 3}{4 \cdot 7} x^{\frac{7}{4}} + C$

Step 7.3.2.4.2.2

Cancel the common factor.

$\frac{1}{x^{\frac{1}{4}}} y = \frac{4 \cdot 3}{4 \cdot 7} x^{\frac{7}{4}} + C$

Step 7.3.2.4.2.3

Rewrite the expression.

$\frac{1}{x^{\frac{1}{4}}} y = \frac{3}{7} x^{\frac{7}{4}} + C$

Step 8

Solve for $y$ .

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Step 8.1

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

$\frac{y}{x^{\frac{1}{4}}} = \frac{3}{7} x^{\frac{7}{4}} + C$

Step 8.2

Combine $\frac{3}{7}$ and $x^{\frac{7}{4}}$ .

$\frac{y}{x^{\frac{1}{4}}} = \frac{3 x^{\frac{7}{4}}}{7} + C$

Step 8.3

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

$\frac{y}{x^{\frac{1}{4}}} x^{\frac{1}{4}} = (\frac{3 x^{\frac{7}{4}}}{7} + C) x^{\frac{1}{4}}$

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

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Step 8.4.1.1.1

Cancel the common factor.

$\frac{y}{x^{\frac{1}{4}}} x^{\frac{1}{4}} = (\frac{3 x^{\frac{7}{4}}}{7} + C) x^{\frac{1}{4}}$

Step 8.4.1.1.2

Rewrite the expression.

$y = (\frac{3 x^{\frac{7}{4}}}{7} + C) x^{\frac{1}{4}}$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(\frac{3 x^{\frac{7}{4}}}{7} + C) x^{\frac{1}{4}}$ .

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Step 8.4.2.1.1

Apply the distributive property.

$y = \frac{3 x^{\frac{7}{4}}}{7} x^{\frac{1}{4}} + C x^{\frac{1}{4}}$

Step 8.4.2.1.2

Multiply $\frac{3 x^{\frac{7}{4}}}{7} x^{\frac{1}{4}}$ .

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Step 8.4.2.1.2.1

Combine $\frac{3 x^{\frac{7}{4}}}{7}$ and $x^{\frac{1}{4}}$ .

$y = \frac{3 x^{\frac{7}{4}} x^{\frac{1}{4}}}{7} + C x^{\frac{1}{4}}$

Step 8.4.2.1.2.2

Multiply $x^{\frac{7}{4}}$ by $x^{\frac{1}{4}}$ by adding the exponents.

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Step 8.4.2.1.2.2.1

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

$y = \frac{3 (x^{\frac{1}{4}} x^{\frac{7}{4}})}{7} + C x^{\frac{1}{4}}$

Step 8.4.2.1.2.2.2

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

$y = \frac{3 x^{\frac{1}{4} + \frac{7}{4}}}{7} + C x^{\frac{1}{4}}$

Step 8.4.2.1.2.2.3

Combine the numerators over the common denominator.

$y = \frac{3 x^{\frac{1 + 7}{4}}}{7} + C x^{\frac{1}{4}}$

Step 8.4.2.1.2.2.4

Add $1$ and $7$ .

$y = \frac{3 x^{\frac{8}{4}}}{7} + C x^{\frac{1}{4}}$

Step 8.4.2.1.2.2.5

Divide $8$ by $4$ .

$y = \frac{3 x^{2}}{7} + C x^{\frac{1}{4}}$