Calculus Examples

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

Step 1

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

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Step 1.1

Divide each term in $y \frac{d x}{d y} - 2 x = 3 y^{2} - 2$ by $y$ .

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

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Cancel the common factor.

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

Step 1.3.2.4

Rewrite the expression.

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

Step 1.3.2.5

Divide $3 y$ by $1$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

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

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Step 2.2.1

Split the fraction into multiple fractions.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $2$ by $- 1$ .

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

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

Step 2.5

Exponentiation and log are inverse functions.

$y^{- 2}$

Step 2.6

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

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

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Step 3.2.5.2.1

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

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Step 3.2.5.2.1.1

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

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

Step 3.2.5.2.1.2

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

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

Step 3.2.5.2.2

Add $1$ and $2$ .

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

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.2

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

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

Step 3.3.3

Cancel the common factor of $y$ .

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Step 3.3.3.1

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

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

Step 3.3.3.2

Cancel the common factor.

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

Step 3.3.3.3

Rewrite the expression.

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

Step 3.3.4

Combine.

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

Step 3.3.5

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

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Step 3.3.5.1

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

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Step 3.3.5.1.1

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

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

Step 3.3.5.1.2

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

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

Step 3.3.5.2

Add $2$ and $1$ .

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

Step 3.3.6

Multiply $- 2$ by $1$ .

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

Step 3.3.7

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

Simplify the expression.

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Step 7.6.1

Multiply $2$ by $- 1$ .

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

Step 7.6.2

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

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

Step 7.6.3

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

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Step 7.6.3.1

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

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

Step 7.6.3.2

Multiply $3$ by $- 1$ .

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

Step 7.7

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

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

Step 7.8

Simplify.

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Step 7.8.1

Simplify.

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Step 7.8.1.1

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

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

Step 7.8.1.2

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

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

Step 7.8.2

Simplify.

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

Step 7.8.3

Simplify.

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Step 7.8.3.1

Move the negative in front of the fraction.

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

Step 7.8.3.2

Multiply $- 1$ by $- 1$ .

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

Step 7.8.3.3

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

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

Step 8

Solve for $x$ .

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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 $3 \ln (| y |)$ from both sides of the equation.

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

Step 8.1.2

Subtract $\frac{1}{y^{2}}$ from both sides of the equation.

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

Step 8.1.3

Subtract $C$ from both sides of the equation.

$\frac{1}{y^{2}} x - 3 \ln (| y |) - \frac{1}{y^{2}} - C = 0$

Step 8.1.4

Simplify each term.

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Step 8.1.4.1

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

$\frac{x}{y^{2}} - 3 \ln (| y |) - \frac{1}{y^{2}} - C = 0$

Step 8.1.4.2

Simplify $- 3 \ln (| y |)$ by moving $3$ inside the logarithm.

$\frac{x}{y^{2}} - \ln ({| y |}^{3}) - \frac{1}{y^{2}} - C = 0$

Step 8.2

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

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Step 8.2.1

Add $\ln ({| y |}^{3})$ to both sides of the equation.

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

Step 8.2.2

Add $\frac{1}{y^{2}}$ to both sides of the equation.

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

Step 8.2.3

Add $C$ to both sides of the equation.

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

Step 8.3

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

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

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 $y^{2}$ .

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Step 8.4.1.1.1

Cancel the common factor.

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

Step 8.4.1.1.2

Rewrite the expression.

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

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(\ln ({| y |}^{3}) + \frac{1}{y^{2}} + C) y^{2}$ .

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Step 8.4.2.1.1

Apply the distributive property.

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

Step 8.4.2.1.2

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

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Step 8.4.2.1.2.1

Cancel the common factor.

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

Step 8.4.2.1.2.2

Rewrite the expression.

$x = \ln ({| y |}^{3}) y^{2} + 1 + C y^{2}$

Step 8.4.2.1.3

Simplify the expression.

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Step 8.4.2.1.3.1

Reorder factors in $\ln ({| y |}^{3}) y^{2} + 1 + C y^{2}$ .

$x = y^{2} \ln ({| y |}^{3}) + 1 + C y^{2}$

Step 8.4.2.1.3.2

Move $1$ .

$x = y^{2} \ln ({| y |}^{3}) + C y^{2} + 1$

Step 8.4.2.1.3.3

Reorder $y^{2} \ln ({| y |}^{3})$ and $C y^{2}$ .

$x = C y^{2} + y^{2} \ln ({| y |}^{3}) + 1$