Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Split the fraction $\frac{3 y - 4 x}{x}$ into two fractions.

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

Step 1.1.2

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

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

Step 1.2

Factor $y$ out of $- \frac{3 y}{x}$ .

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

Step 1.3

Reorder $y$ and $- \frac{3}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - \frac{3}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int - \frac{3}{x} d x}$

Step 2.2

Integrate $- \frac{3}{x}$ .

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Step 2.2.1

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

$e^{- \int \frac{3}{x} d x}$

Step 2.2.2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{- (3 \int \frac{1}{x} d x)}$

Step 2.2.3

Multiply $3$ by $- 1$ .

$e^{- 3 \int \frac{1}{x} d x}$

Step 2.2.4

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

$e^{- 3 (\ln (| x |) + C)}$

Step 2.2.5

Simplify.

$e^{- 3 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- 3 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 3})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 3}$

Step 2.6

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

$\frac{1}{x^{3}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{3}}$ .

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Step 3.1

Multiply each term by $\frac{1}{x^{3}}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $- \frac{1}{x^{3}} \cdot \frac{3 y}{x}$ .

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Step 3.2.4.1

Multiply $\frac{3 y}{x}$ by $\frac{1}{x^{3}}$ .

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

Step 3.2.4.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 3.2.4.2.1

Multiply $x$ by $x^{3}$ .

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Step 3.2.4.2.1.1

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

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

Step 3.2.4.2.1.2

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

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

Step 3.2.4.2.2

Add $1$ and $3$ .

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

Step 3.3

Combine.

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

Step 3.4

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 3.4.1

Multiply $x^{3}$ by $x$ .

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Step 3.4.1.1

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

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

Step 3.4.1.2

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

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

Step 3.4.2

Add $3$ and $1$ .

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

Step 3.5

Multiply $- 4 x$ by $1$ .

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

Step 3.6

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

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Step 3.6.1

Factor $x$ out of $- 4 x$ .

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

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

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

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

Step 3.6.2.2

Cancel the common factor.

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

Step 3.6.2.3

Rewrite the expression.

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

Step 3.7

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

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.

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

Step 7.2

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

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

Step 7.3

Simplify the expression.

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Step 7.3.1

Multiply $4$ by $- 1$ .

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

Step 7.3.2

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

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

Step 7.3.3

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

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Step 7.3.3.1

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

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

Step 7.3.3.2

Multiply $3$ by $- 1$ .

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

Step 7.4

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

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

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

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

Step 7.5.1.2

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

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

Step 7.5.2

Simplify.

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

Step 7.5.3

Simplify.

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Step 7.5.3.1

Multiply $- 1$ by $- 4$ .

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

Step 7.5.3.2

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

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

Step 7.5.3.3

Cancel the common factor of $4$ and $2$ .

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Step 7.5.3.3.1

Factor $2$ out of $4$ .

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

Step 7.5.3.3.2

Cancel the common factors.

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Step 7.5.3.3.2.1

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

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

Step 7.5.3.3.2.2

Cancel the common factor.

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

Step 7.5.3.3.2.3

Rewrite the expression.

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

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

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

Step 8.1.2

Subtract $C$ from both sides of the equation.

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

Step 8.1.3

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

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

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

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

Step 8.2.2

Add $C$ to both sides of the equation.

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

Step 8.3

Multiply both sides by $x^{3}$ .

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

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 $x^{3}$ .

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Step 8.4.1.1.1

Cancel the common factor.

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

Step 8.4.1.1.2

Rewrite the expression.

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

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

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

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Step 8.4.2.1.1

Apply the distributive property.

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

Step 8.4.2.1.2

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

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Step 8.4.2.1.2.1

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

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

Step 8.4.2.1.2.2

Cancel the common factor.

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

Step 8.4.2.1.2.3

Rewrite the expression.

$y = 2 x + C x^{3}$

Step 8.4.2.1.3

Reorder $2 x$ and $C x^{3}$ .

$y = C x^{3} + 2 x$