Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [6 x^{2} y]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 0 + 6 x^{2} \frac{d}{d y} [y]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 0 + 6 x^{2} \cdot 1$

Step 1.3.3

Multiply $6$ by $1$ .

$\frac{\partial M}{\partial y} = 0 + 6 x^{2}$

Step 1.4

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

$\frac{\partial M}{\partial y} = 6 x^{2}$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

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Step 2.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x^{3}]$ .

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

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 2.3.3

Multiply $3$ by $3$ .

$\frac{\partial N}{\partial x} = 9 x^{2} + \frac{d}{d x} [- 2 x y]$

Step 2.4

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

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Step 2.4.1

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

$\frac{\partial N}{\partial x} = 9 x^{2} - 2 y \frac{d}{d x} [x]$

Step 2.4.2

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

$\frac{\partial N}{\partial x} = 9 x^{2} - 2 y \cdot 1$

Step 2.4.3

Multiply $- 2$ by $1$ .

$\frac{\partial N}{\partial x} = 9 x^{2} - 2 y$

Step 3

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

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Step 3.1

Substitute $6 x^{2}$ for $\frac{\partial M}{\partial y}$ and $9 x^{2} - 2 y$ for $\frac{\partial N}{\partial x}$ .

$6 x^{2} = 9 x^{2} - 2 y$

Step 3.2

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

$6 x^{2} = 9 x^{2} - 2 y$ is not an identity.

Step 4

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

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Step 4.1

Substitute $6 x^{2}$ for $\frac{\partial M}{\partial y}$ .

$\frac{6 x^{2} - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $9 x^{2} - 2 y$ for $\frac{\partial N}{\partial x}$ .

$\frac{6 x^{2} - (9 x^{2} - 2 y)}{N}$

Step 4.3

Substitute $3 x^{3} - 2 x y$ for $N$ .

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Step 4.3.1

Substitute $3 x^{3} - 2 x y$ for $N$ .

$\frac{6 x^{2} - (9 x^{2} - 2 y)}{3 x^{3} - 2 x y}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

$\frac{6 x^{2} - (9 x^{2}) - (- 2 y)}{3 x^{3} - 2 x y}$

Step 4.3.2.2

Multiply $9$ by $- 1$ .

$\frac{6 x^{2} - 9 x^{2} - (- 2 y)}{3 x^{3} - 2 x y}$

Step 4.3.2.3

Multiply $- 2$ by $- 1$ .

$\frac{6 x^{2} - 9 x^{2} + 2 y}{3 x^{3} - 2 x y}$

Step 4.3.2.4

Subtract $9 x^{2}$ from $6 x^{2}$ .

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

Step 4.3.3

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

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Step 4.3.3.1

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

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

Step 4.3.3.2

Factor $x$ out of $- 2 x y$ .

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

Step 4.3.3.3

Factor $x$ out of $x (3 x^{2}) + x (- 2 y)$ .

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

Step 4.3.4

Cancel the common factor of $- 3 x^{2} + 2 y$ and $3 x^{2} - 2 y$ .

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Step 4.3.4.1

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 4.3.4.2

Factor $- 1$ out of $2 y$ .

$\frac{- (3 x^{2}) - (- 2 y)}{x (3 x^{2} - 2 y)}$

Step 4.3.4.3

Factor $- 1$ out of $- (3 x^{2}) - (- 2 y)$ .

$\frac{- (3 x^{2} - 2 y)}{x (3 x^{2} - 2 y)}$

Step 4.3.4.4

Rewrite $- (3 x^{2} - 2 y)$ as $- 1 (3 x^{2} - 2 y)$ .

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

Step 4.3.4.5

Cancel the common factor.

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

Step 4.3.4.6

Rewrite the expression.

$\frac{- 1}{x}$

Step 4.3.5

Move the negative in front of the fraction.

$- \frac{1}{x}$

Step 4.4

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

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

Step 5

Evaluate the integral $e^{\int - \frac{1}{x} d x}$ .

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Step 5.1

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

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

Step 5.2

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

$μ (x, y) = e^{- (\ln (| x |) + C)}$

Step 5.3

Simplify.

$μ (x, y) = e^{- \ln (| x |)} + C$

Step 5.4

Simplify each term.

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Step 5.4.1

Simplify $- \ln (| x |)$ by moving $- 1$ inside the logarithm.

$μ (x, y) = e^{\ln ({| x |}^{- 1})} + C$

Step 5.4.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| x |}^{- 1} + C$

Step 5.4.3

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

$μ (x, y) = \frac{1}{| x |} + C$

Step 6

Multiply both sides of $(2 x + 6 x^{2} y) d x + (3 x^{3} - 2 x y) d y = 0$ by the integration factor $\frac{1}{x}$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Factor $2 x$ out of $2 x + 6 x^{2} y$ .

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Step 6.3.1

Factor $2 x$ out of $2 x$ .

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

Step 6.3.2

Factor $2 x$ out of $6 x^{2} y$ .

$\frac{2 x (1) + 2 x (3 x y)}{x} d x + (3 x^{3} - 2 x y) d y = 0$

Step 6.3.3

Factor $2 x$ out of $2 x (1) + 2 x (3 x y)$ .

$\frac{2 x (1 + 3 x y)}{x} d x + (3 x^{3} - 2 x y) d y = 0$

Step 6.4

Cancel the common factor of $x$ .

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Step 6.4.1

Cancel the common factor.

$\frac{2 x (1 + 3 x y)}{x} d x + (3 x^{3} - 2 x y) d y = 0$

Step 6.4.2

Divide $2 (1 + 3 x y)$ by $1$ .

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

Step 6.5

Apply the distributive property.

$(2 \cdot 1 + 2 (3 x y)) d x + (3 x^{3} - 2 x y) d y = 0$

Step 6.6

Multiply $2$ by $1$ .

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

Step 6.7

Multiply $3$ by $2$ .

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

Step 6.8

Multiply $3 x^{3} - 2 x y$ by $\frac{1}{x}$ .

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

Step 6.9

Multiply $3 x^{3} - 2 x y$ by $\frac{1}{x}$ .

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

Step 6.10

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

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Step 6.10.1

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

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

Step 6.10.2

Factor $x$ out of $- 2 x y$ .

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

Step 6.10.3

Factor $x$ out of $x (3 x^{2}) + x (- 2 y)$ .

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

Step 6.11

Cancel the common factor of $x$ .

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Step 6.11.1

Cancel the common factor.

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

Step 6.11.2

Divide $3 x^{2} - 2 y$ by $1$ .

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

Step 7

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

$f (x, y) = \int 2 + 6 x y d x$

Step 8

Integrate $M (x, y) = 2 + 6 x y$ to find $f (x, y)$ .

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Step 8.1

Split the single integral into multiple integrals.

$f (x, y) = \int 2 d x + \int 6 x y d x$

Step 8.2

Apply the constant rule.

$f (x, y) = 2 x + C + \int 6 x y d x$

Step 8.3

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

$f (x, y) = 2 x + C + 6 y \int x d x$

Step 8.4

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

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

Step 8.5

Simplify.

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

Step 8.6

Simplify.

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Step 8.6.1

Combine $6$ and $\frac{1}{2}$ .

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

Step 8.6.2

Cancel the common factor of $6$ and $2$ .

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Step 8.6.2.1

Factor $2$ out of $6$ .

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

Step 8.6.2.2

Cancel the common factors.

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Step 8.6.2.2.1

Factor $2$ out of $2$ .

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

Step 8.6.2.2.2

Cancel the common factor.

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

Step 8.6.2.2.3

Rewrite the expression.

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

Step 8.6.2.2.4

Divide $3$ by $1$ .

$f (x, y) = 2 x + 3 y x^{2} + C$

Step 9

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

$f (x, y) = 2 x + 3 y x^{2} + g (y)$

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

Differentiate.

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Step 11.2.1

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

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

Step 11.2.2

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

$0 + \frac{d}{d y} [3 y x^{2}] + \frac{d}{d y} [g (y)] = 3 x^{2} - 2 y$

Step 11.3

Evaluate $\frac{d}{d y} [3 y x^{2}]$ .

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Step 11.3.1

Since $3 x^{2}$ is constant with respect to $y$ , the derivative of $3 y x^{2}$ with respect to $y$ is $3 x^{2} \frac{d}{d y} [y]$ .

$0 + 3 x^{2} \frac{d}{d y} [y] + \frac{d}{d y} [g (y)] = 3 x^{2} - 2 y$

Step 11.3.2

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

$0 + 3 x^{2} \cdot 1 + \frac{d}{d y} [g (y)] = 3 x^{2} - 2 y$

Step 11.3.3

Multiply $3$ by $1$ .

$0 + 3 x^{2} + \frac{d}{d y} [g (y)] = 3 x^{2} - 2 y$

Step 11.4

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

$0 + 3 x^{2} + \frac{d g}{d y} = 3 x^{2} - 2 y$

Step 11.5

Simplify.

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Step 11.5.1

Add $0$ and $3 x^{2}$ .

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

Step 11.5.2

Reorder terms.

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

Subtract $3 x^{2}$ from both sides of the equation.

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

Step 12.1.2

Combine the opposite terms in $3 x^{2} - 2 y - 3 x^{2}$ .

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Step 12.1.2.1

Subtract $3 x^{2}$ from $3 x^{2}$ .

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

Step 12.1.2.2

Add $- 2 y$ and $0$ .

$\frac{d g}{d y} = - 2 y$

Step 13

Find the antiderivative of $- 2 y$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = - 2 y$ .

$\int \frac{d g}{d y} d y = \int - 2 y d y$

Step 13.2

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

$g (y) = \int - 2 y d y$

Step 13.3

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

$g (y) = - 2 \int y d y$

Step 13.4

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

$g (y) = - 2 (\frac{1}{2} y^{2} + C)$

Step 13.5

Simplify the answer.

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Step 13.5.1

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

$g (y) = - 2 (\frac{1}{2}) y^{2} + C$

Step 13.5.2

Simplify.

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Step 13.5.2.1

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

$g (y) = \frac{- 2}{2} y^{2} + C$

Step 13.5.2.2

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

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Step 13.5.2.2.1

Factor $2$ out of $- 2$ .

$g (y) = \frac{2 \cdot - 1}{2} y^{2} + C$

Step 13.5.2.2.2

Cancel the common factors.

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Step 13.5.2.2.2.1

Factor $2$ out of $2$ .

$g (y) = \frac{2 \cdot - 1}{2 (1)} y^{2} + C$

Step 13.5.2.2.2.2

Cancel the common factor.

$g (y) = \frac{2 \cdot - 1}{2 \cdot 1} y^{2} + C$

Step 13.5.2.2.2.3

Rewrite the expression.

$g (y) = \frac{- 1}{1} y^{2} + C$

Step 13.5.2.2.2.4

Divide $- 1$ by $1$ .

$g (y) = - y^{2} + C$

Step 14

Substitute for $g (y)$ in $f (x, y) = 2 x + 3 y x^{2} + g (y)$ .

$f (x, y) = 2 x + 3 y x^{2} - y^{2} + C$