Calculus Examples

$y d x - (x + 6 y^{2}) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = y$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [y]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$\frac{\partial M}{\partial y} = 1$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = - (x + 6 y^{2})$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [- (x + 6 y^{2})]$

Step 2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- (x + 6 y^{2})$ with respect to $x$ is $- \frac{d}{d x} [x + 6 y^{2}]$ .

$\frac{\partial N}{\partial x} = - \frac{d}{d x} [x + 6 y^{2}]$

Step 2.3

By the Sum Rule, the derivative of $x + 6 y^{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [6 y^{2}]$ .

$\frac{\partial N}{\partial x} = - (\frac{d}{d x} [x] + \frac{d}{d x} [6 y^{2}])$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{\partial N}{\partial x} = - (1 + \frac{d}{d x} [6 y^{2}])$

Step 2.5

Since $6 y^{2}$ is constant with respect to $x$ , the derivative of $6 y^{2}$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = - (1 + 0)$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $1$ and $0$ .

$\frac{\partial N}{\partial x} = - 1 \cdot 1$

Step 2.6.2

Multiply $- 1$ by $1$ .

$\frac{\partial N}{\partial x} = - 1$

Step 3

Check that $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

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Step 3.1

Substitute $1$ for $\frac{\partial M}{\partial y}$ and $- 1$ for $\frac{\partial N}{\partial x}$ .

$1 = - 1$

Step 3.2

Since the left side does not equal the right side, the equation is not an identity.

$1 = - 1$ is not an identity.

Step 4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} d y}$ .

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Step 4.1

Substitute $- 1$ for $\frac{\partial N}{\partial x}$ .

$\frac{- 1 - \frac{\partial M}{\partial y}}{M}$

Step 4.2

Substitute $1$ for $\frac{\partial M}{\partial y}$ .

$\frac{- 1 - 1}{M}$

Step 4.3

Substitute $y$ for $M$ .

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Step 4.3.1

Substitute $y$ for $M$ .

$\frac{- 1 - 1}{y}$

Step 4.3.2

Subtract $1$ from $- 1$ .

$\frac{- 2}{y}$

Step 4.3.3

Substitute $y$ for $M$ .

$- \frac{2}{y}$

Step 4.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} d y}$ .

$μ (x, y) = e^{\int - \frac{2}{y} d y}$

Step 5

Evaluate the integral $e^{\int - \frac{2}{y} d y}$ .

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Step 5.1

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

$μ (x, y) = e^{- \int \frac{2}{y} d y}$

Step 5.2

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

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

Step 5.3

Multiply $2$ by $- 1$ .

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

Step 5.4

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

$μ (x, y) = e^{- 2 (\ln (| y |) + C)}$

Step 5.5

Simplify.

$μ (x, y) = e^{- 2 \ln (| y |)} + C$

Step 5.6

Simplify each term.

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Step 5.6.1

Simplify $- 2 \ln (| y |)$ by moving $- 2$ inside the logarithm.

$μ (x, y) = e^{\ln ({| y |}^{- 2})} + C$

Step 5.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| y |}^{- 2} + C$

Step 5.6.3

Remove the absolute value in ${| y |}^{- 2}$ because exponentiations with even powers are always positive.

$μ (x, y) = y^{- 2} + C$

Step 5.6.4

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

$μ (x, y) = \frac{1}{y^{2}} + C$

Step 6

Multiply both sides of $y d x - (x + 6 y^{2}) d y = 0$ by the integration factor $\frac{1}{y^{2}}$ .

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Step 6.1

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

$y \frac{1}{y^{2}} d x - (x + 6 y^{2}) d y = 0$

Step 6.2

Cancel the common factor of $y$ .

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Step 6.2.1

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

$y \frac{1}{y \cdot y} d x - (x + 6 y^{2}) d y = 0$

Step 6.2.2

Cancel the common factor.

$y \frac{1}{y \cdot y} d x - (x + 6 y^{2}) d y = 0$

Step 6.2.3

Rewrite the expression.

$\frac{1}{y} d x - (x + 6 y^{2}) d y = 0$

Step 6.3

Multiply $- (x + 6 y^{2})$ by $\frac{1}{y^{2}}$ .

$\frac{1}{y} d x - (x + 6 y^{2}) \frac{1}{y^{2}} d y = 0$

Step 6.4

Apply the distributive property.

$\frac{1}{y} d x + (- x - (6 y^{2})) \frac{1}{y^{2}} d y = 0$

Step 6.5

Multiply $6$ by $- 1$ .

$\frac{1}{y} d x + (- x - 6 y^{2}) \frac{1}{y^{2}} d y = 0$

Step 6.6

Multiply $- x - 6 y^{2}$ by $\frac{1}{y^{2}}$ .

$\frac{1}{y} d x + \frac{- x - 6 y^{2}}{y^{2}} d y = 0$

Step 6.7

Factor $- 1$ out of $- x$ .

$\frac{1}{y} d x + \frac{- (x) - 6 y^{2}}{y^{2}} d y = 0$

Step 6.8

Factor $- 1$ out of $- 6 y^{2}$ .

$\frac{1}{y} d x + \frac{- (x) - (6 y^{2})}{y^{2}} d y = 0$

Step 6.9

Factor $- 1$ out of $- (x) - (6 y^{2})$ .

$\frac{1}{y} d x + \frac{- (x + 6 y^{2})}{y^{2}} d y = 0$

Step 6.10

Rewrite $- (x + 6 y^{2})$ as $- 1 (x + 6 y^{2})$ .

$\frac{1}{y} d x + \frac{- 1 (x + 6 y^{2})}{y^{2}} d y = 0$

Step 6.11

Move the negative in front of the fraction.

$\frac{1}{y} d x - \frac{x + 6 y^{2}}{y^{2}} d y = 0$

Step 7

Set $f (x, y)$ equal to the integral of $M (x, y)$ .

$f (x, y) = \int \frac{1}{y} d x$

Step 8

Integrate $M (x, y) = \frac{1}{y}$ to find $f (x, y)$ .

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Step 8.1

Apply the constant rule.

$f (x, y) = \frac{1}{y} x + C$

Step 8.2

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

$f (x, y) = \frac{x}{y} + C$

Step 9

Since the integral of $g (y)$ will contain an integration constant, we can replace $C$ with $g (y)$ .

$f (x, y) = \frac{x}{y} + g (y)$

Step 10

Set $\frac{\partial f}{\partial y} = N (x, y)$ .

$\frac{\partial f}{\partial y} = - \frac{x + 6 y^{2}}{y^{2}}$

Step 11

Find $\frac{\partial f}{\partial y}$ .

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Step 11.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [\frac{x}{y} + g (y)] = - \frac{x + 6 y^{2}}{y^{2}}$

Step 11.2

By the Sum Rule, the derivative of $\frac{x}{y} + g (y)$ with respect to $y$ is $\frac{d}{d y} [\frac{x}{y}] + \frac{d}{d y} [g (y)]$ .

$\frac{d}{d y} [\frac{x}{y}] + \frac{d}{d y} [g (y)] = - \frac{x + 6 y^{2}}{y^{2}}$

Step 11.3

Evaluate $\frac{d}{d y} [\frac{x}{y}]$ .

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Step 11.3.1

Since $x$ is constant with respect to $y$ , the derivative of $\frac{x}{y}$ with respect to $y$ is $x \frac{d}{d y} [\frac{1}{y}]$ .

$x \frac{d}{d y} [\frac{1}{y}] + \frac{d}{d y} [g (y)] = - \frac{x + 6 y^{2}}{y^{2}}$

Step 11.3.2

Rewrite $\frac{1}{y}$ as $y^{- 1}$ .

$x \frac{d}{d y} [y^{- 1}] + \frac{d}{d y} [g (y)] = - \frac{x + 6 y^{2}}{y^{2}}$

Step 11.3.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = - 1$ .

$x (- y^{- 2}) + \frac{d}{d y} [g (y)] = - \frac{x + 6 y^{2}}{y^{2}}$

Step 11.4

Differentiate using the function rule which states that the derivative of $g (y)$ is $\frac{d g}{d y}$ .

$x (- y^{- 2}) + \frac{d g}{d y} = - \frac{x + 6 y^{2}}{y^{2}}$

Step 11.5

Simplify.

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Step 11.5.1

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

$x (- \frac{1}{y^{2}}) + \frac{d g}{d y} = - \frac{x + 6 y^{2}}{y^{2}}$

Step 11.5.2

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

$- \frac{x}{y^{2}} + \frac{d g}{d y} = - \frac{x + 6 y^{2}}{y^{2}}$

Step 11.5.3

Reorder terms.

$\frac{d g}{d y} - \frac{x}{y^{2}} = - \frac{x + 6 y^{2}}{y^{2}}$

Step 12

Solve for $\frac{d g}{d y}$ .

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Step 12.1

Solve for $\frac{d g}{d y}$ .

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Step 12.1.1

Move all terms containing variables to the left side of the equation.

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Step 12.1.1.1

Add $\frac{x + 6 y^{2}}{y^{2}}$ to both sides of the equation.

$\frac{d g}{d y} - \frac{x}{y^{2}} + \frac{x + 6 y^{2}}{y^{2}} = 0$

Step 12.1.1.2

Combine the numerators over the common denominator.

$\frac{d g}{d y} + \frac{- x + x + 6 y^{2}}{y^{2}} = 0$

Step 12.1.1.3

Add $- x$ and $x$ .

$\frac{d g}{d y} + \frac{0 + 6 y^{2}}{y^{2}} = 0$

Step 12.1.1.4

Add $0$ and $6 y^{2}$ .

$\frac{d g}{d y} + \frac{6 y^{2}}{y^{2}} = 0$

Step 12.1.1.5

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

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Step 12.1.1.5.1

Cancel the common factor.

$\frac{d g}{d y} + \frac{6 y^{2}}{y^{2}} = 0$

Step 12.1.1.5.2

Divide $6$ by $1$ .

$\frac{d g}{d y} + 6 = 0$

Step 12.1.2

Subtract $6$ from both sides of the equation.

$\frac{d g}{d y} = - 6$

Step 13

Find the antiderivative of $- 6$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = - 6$ .

$\int \frac{d g}{d y} d y = \int - 6 d y$

Step 13.2

Evaluate $\int \frac{d g}{d y} d y$ .

$g (y) = \int - 6 d y$

Step 13.3

Apply the constant rule.

$g (y) = - 6 y + C$

Step 14

Substitute for $g (y)$ in $f (x, y) = \frac{x}{y} + g (y)$ .

$f (x, y) = \frac{x}{y} - 6 y + C$