Calculus Examples

$2 x y d x + (6 x^{2} + y^{3}) d y = 0$

Step 1

Write the problem as a mathematical expression.

$2 x y d x + (6 x^{2} + y^{3}) d y = 0$

Step 2

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 2 x y$ .

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Step 2.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [2 x y]$

Step 2.2

Since $2 x$ is constant with respect to $y$ , the derivative of $2 x y$ with respect to $y$ is $2 x \frac{d}{d y} [y]$ .

$\frac{\partial M}{\partial y} = 2 x \frac{d}{d y} [y]$

Step 2.3

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

$\frac{\partial M}{\partial y} = 2 x \cdot 1$

Step 2.4

Multiply $2$ by $1$ .

$\frac{\partial M}{\partial y} = 2 x$

Step 3

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = 6 x^{2} + y^{3}$ .

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Step 3.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [6 x^{2} + y^{3}]$

Step 3.2

By the Sum Rule, the derivative of $6 x^{2} + y^{3}$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [y^{3}]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [y^{3}]$

Step 3.3

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 3.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$\frac{\partial N}{\partial x} = 6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{3}]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{\partial N}{\partial x} = 6 (2 x) + \frac{d}{d x} [y^{3}]$

Step 3.3.3

Multiply $2$ by $6$ .

$\frac{\partial N}{\partial x} = 12 x + \frac{d}{d x} [y^{3}]$

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $y^{3}$ is constant with respect to $x$ , the derivative of $y^{3}$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = 12 x + 0$

Step 3.4.2

Add $12 x$ and $0$ .

$\frac{\partial N}{\partial x} = 12 x$

Step 4

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

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Step 4.1

Substitute $2 x$ for $\frac{\partial M}{\partial y}$ and $12 x$ for $\frac{\partial N}{\partial x}$ .

$2 x = 12 x$

Step 4.2

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

$2 x = 12 x$ is not an identity.

Step 5

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 5.1

Substitute $12 x$ for $\frac{\partial N}{\partial x}$ .

$\frac{12 x - \frac{\partial M}{\partial y}}{M}$

Step 5.2

Substitute $2 x$ for $\frac{\partial M}{\partial y}$ .

$\frac{12 x - (2 x)}{M}$

Step 5.3

Substitute $2 x y$ for $M$ .

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Step 5.3.1

Substitute $2 x y$ for $M$ .

$\frac{12 x - (2 x)}{2 x y}$

Step 5.3.2

Simplify the numerator.

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Step 5.3.2.1

Factor $x$ out of $12 x - 1 \cdot 2 x$ .

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Step 5.3.2.1.1

Factor $x$ out of $12 x$ .

$\frac{x \cdot 12 - 1 \cdot 2 x}{2 x y}$

Step 5.3.2.1.2

Factor $x$ out of $- 1 \cdot 2 x$ .

$\frac{x \cdot 12 + x (- 1 \cdot 2)}{2 x y}$

Step 5.3.2.1.3

Factor $x$ out of $x \cdot 12 + x (- 1 \cdot 2)$ .

$\frac{x (12 - 1 \cdot 2)}{2 x y}$

Step 5.3.2.2

Multiply $- 1$ by $2$ .

$\frac{x (12 - 2)}{2 x y}$

Step 5.3.2.3

Subtract $2$ from $12$ .

$\frac{x \cdot 10}{2 x y}$

Step 5.3.3

Cancel the common factor of $x$ .

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Step 5.3.3.1

Cancel the common factor.

$\frac{x \cdot 10}{2 x y}$

Step 5.3.3.2

Rewrite the expression.

$\frac{10}{2 y}$

Step 5.3.4

Substitute $2 x y$ for $M$ .

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Step 5.3.4.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 5}{2 y}$

Step 5.3.4.2

Cancel the common factors.

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Step 5.3.4.2.1

Factor $2$ out of $2 y$ .

$\frac{2 \cdot 5}{2 (y)}$

Step 5.3.4.2.2

Cancel the common factor.

$\frac{2 \cdot 5}{2 y}$

Step 5.3.4.2.3

Rewrite the expression.

$\frac{5}{y}$

Step 5.4

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

$μ (x, y) = e^{\int \frac{5}{y} d y}$

Step 6

Evaluate the integral $e^{\int \frac{5}{y} d y}$ .

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Step 6.1

Since $5$ is constant with respect to $y$ , move $5$ out of the integral.

$μ (x, y) = e^{5 \int \frac{1}{y} d y}$

Step 6.2

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

$μ (x, y) = e^{5 (\ln (| y |) + C)}$

Step 6.3

Simplify.

$μ (x, y) = e^{5 \ln (| y |)} + C$

Step 6.4

Simplify each term.

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Step 6.4.1

Simplify $5 \ln (| y |)$ by moving $5$ inside the logarithm.

$μ (x, y) = e^{\ln ({| y |}^{5})} + C$

Step 6.4.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| y |}^{5} + C$

Step 7

Multiply both sides of $2 x y d x + (6 x^{2} + y^{3}) d y = 0$ by the integration factor $y^{5}$ .

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Step 7.1

Multiply $2 x y$ by $y^{5}$ .

$2 x y \cdot y^{5} d x + (6 x^{2} + y^{3}) d y = 0$

Step 7.2

Multiply $y$ by $y^{5}$ by adding the exponents.

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Step 7.2.1

Move $y^{5}$ .

$2 x (y^{5} y) d x + (6 x^{2} + y^{3}) d y = 0$

Step 7.2.2

Multiply $y^{5}$ by $y$ .

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Step 7.2.2.1

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

$2 x (y^{5} y^{1}) d x + (6 x^{2} + y^{3}) d y = 0$

Step 7.2.2.2

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

$2 x y^{5 + 1} d x + (6 x^{2} + y^{3}) d y = 0$

Step 7.2.3

Add $5$ and $1$ .

$2 x y^{6} d x + (6 x^{2} + y^{3}) d y = 0$

Step 7.3

Multiply $6 x^{2} + y^{3}$ by $y^{5}$ .

$2 x y^{6} d x + (6 x^{2} + y^{3}) y^{5} d y = 0$

Step 7.4

Apply the distributive property.

$2 x y^{6} d x + (6 x^{2} y^{5} + y^{3} y^{5}) d y = 0$

Step 7.5

Multiply $y^{3}$ by $y^{5}$ by adding the exponents.

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Step 7.5.1

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

$2 x y^{6} d x + (6 x^{2} y^{5} + y^{3 + 5}) d y = 0$

Step 7.5.2

Add $3$ and $5$ .

$2 x y^{6} d x + (6 x^{2} y^{5} + y^{8}) d y = 0$

Step 8

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

$f (x, y) = \int 2 x y^{6} d x$

Step 9

Integrate $M (x, y) = 2 x y^{6}$ to find $f (x, y)$ .

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Step 9.1

Since $2 y^{6}$ is constant with respect to $x$ , move $2 y^{6}$ out of the integral.

$f (x, y) = 2 y^{6} \int x d x$

Step 9.2

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

$f (x, y) = 2 y^{6} (\frac{1}{2} x^{2} + C)$

Step 9.3

Simplify the answer.

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Step 9.3.1

Rewrite $2 y^{6} (\frac{1}{2} x^{2} + C)$ as $2 y^{6} \frac{1}{2} x^{2} + C$ .

$f (x, y) = 2 y^{6} \frac{1}{2} x^{2} + C$

Step 9.3.2

Simplify.

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Step 9.3.2.1

Combine $2$ and $\frac{1}{2}$ .

$f (x, y) = y^{6} \frac{2}{2} x^{2} + C$

Step 9.3.2.2

Combine $y^{6}$ and $\frac{2}{2}$ .

$f (x, y) = \frac{y^{6} \cdot 2}{2} x^{2} + C$

Step 9.3.2.3

Move $2$ to the left of $y^{6}$ .

$f (x, y) = \frac{2 \cdot y^{6}}{2} x^{2} + C$

Step 9.3.2.4

Multiply $2$ by $y^{6}$ .

$f (x, y) = \frac{2 y^{6}}{2} x^{2} + C$

Step 9.3.2.5

Cancel the common factor of $2$ .

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Step 9.3.2.5.1

Cancel the common factor.

$f (x, y) = \frac{2 y^{6}}{2} x^{2} + C$

Step 9.3.2.5.2

Divide $y^{6}$ by $1$ .

$f (x, y) = y^{6} x^{2} + C$

Step 10

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

$f (x, y) = y^{6} x^{2} + g (y)$

Step 11

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

$\frac{\partial f}{\partial y} = 6 x^{2} y^{5} + y^{8}$

Step 12

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

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Step 12.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [y^{6} x^{2} + g (y)] = 6 x^{2} y^{5} + y^{8}$

Step 12.2

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

$\frac{d}{d y} [y^{6} x^{2}] + \frac{d}{d y} [g (y)] = 6 x^{2} y^{5} + y^{8}$

Step 12.3

Evaluate $\frac{d}{d y} [y^{6} x^{2}]$ .

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Step 12.3.1

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

$x^{2} \frac{d}{d y} [y^{6}] + \frac{d}{d y} [g (y)] = 6 x^{2} y^{5} + y^{8}$

Step 12.3.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 6$ .

$x^{2} (6 y^{5}) + \frac{d}{d y} [g (y)] = 6 x^{2} y^{5} + y^{8}$

Step 12.3.3

Move $6$ to the left of $x^{2}$ .

$6 x^{2} y^{5} + \frac{d}{d y} [g (y)] = 6 x^{2} y^{5} + y^{8}$

Step 12.4

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

$6 x^{2} y^{5} + \frac{d g}{d y} = 6 x^{2} y^{5} + y^{8}$

Step 12.5

Reorder terms.

$\frac{d g}{d y} + 6 y^{5} x^{2} = 6 x^{2} y^{5} + y^{8}$

Step 13

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

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Step 13.1

Move all terms not containing $\frac{d g}{d y}$ to the right side of the equation.

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Step 13.1.1

Subtract $6 y^{5} x^{2}$ from both sides of the equation.

$\frac{d g}{d y} = 6 x^{2} y^{5} + y^{8} - 6 y^{5} x^{2}$

Step 13.1.2

Combine the opposite terms in $6 x^{2} y^{5} + y^{8} - 6 y^{5} x^{2}$ .

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Step 13.1.2.1

Reorder the factors in the terms $6 x^{2} y^{5}$ and $- 6 y^{5} x^{2}$ .

$\frac{d g}{d y} = 6 y^{5} x^{2} + y^{8} - 6 y^{5} x^{2}$

Step 13.1.2.2

Subtract $6 y^{5} x^{2}$ from $6 y^{5} x^{2}$ .

$\frac{d g}{d y} = y^{8} + 0$

Step 13.1.2.3

Add $y^{8}$ and $0$ .

$\frac{d g}{d y} = y^{8}$

Step 14

Find the antiderivative of $y^{8}$ to find $g (y)$ .

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Step 14.1

Integrate both sides of $\frac{d g}{d y} = y^{8}$ .

$\int \frac{d g}{d y} d y = \int y^{8} d y$

Step 14.2

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

$g (y) = \int y^{8} d y$

Step 14.3

By the Power Rule, the integral of $y^{8}$ with respect to $y$ is $\frac{1}{9} y^{9}$ .

$g (y) = \frac{1}{9} y^{9} + C$

Step 15

Substitute for $g (y)$ in $f (x, y) = y^{6} x^{2} + g (y)$ .

$f (x, y) = y^{6} x^{2} + \frac{1}{9} y^{9} + C$

Step 16

Combine $\frac{1}{9}$ and $y^{9}$ .

$f (x, y) = y^{6} x^{2} + \frac{y^{9}}{9} + C$