Calculus Examples

$(x^{2} y^{4} - 1) d x + x^{3} y^{3} d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = x^{2} y^{4} - 1$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [x^{2} y^{4} - 1]$

Step 1.2

By the Sum Rule, the derivative of $x^{2} y^{4} - 1$ with respect to $y$ is $\frac{d}{d y} [x^{2} y^{4}] + \frac{d}{d y} [- 1]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [x^{2} y^{4}] + \frac{d}{d y} [- 1]$

Step 1.3

Evaluate $\frac{d}{d y} [x^{2} y^{4}]$ .

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Step 1.3.1

Since $x^{2}$ is constant with respect to $y$ , the derivative of $x^{2} y^{4}$ with respect to $y$ is $x^{2} \frac{d}{d y} [y^{4}]$ .

$\frac{\partial M}{\partial y} = x^{2} \frac{d}{d y} [y^{4}] + \frac{d}{d y} [- 1]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 4$ .

$\frac{\partial M}{\partial y} = x^{2} (4 y^{3}) + \frac{d}{d y} [- 1]$

Step 1.3.3

Move $4$ to the left of $x^{2}$ .

$\frac{\partial M}{\partial y} = 4 x^{2} y^{3} + \frac{d}{d y} [- 1]$

Step 1.4

Since $- 1$ is constant with respect to $y$ , the derivative of $- 1$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 4 x^{2} y^{3} + 0$

Step 1.5

Simplify.

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Step 1.5.1

Add $4 x^{2} y^{3}$ and $0$ .

$\frac{\partial M}{\partial y} = 4 x^{2} y^{3}$

Step 1.5.2

Reorder the factors of $4 x^{2} y^{3}$ .

$\frac{\partial M}{\partial y} = 4 y^{3} x^{2}$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

$\frac{\partial N}{\partial x} = y^{3} (3 x^{2})$

Step 2.4

Move $3$ to the left of $y^{3}$ .

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

Step 3

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

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Step 3.1

Substitute $4 y^{3} x^{2}$ for $\frac{\partial M}{\partial y}$ and $3 y^{3} x^{2}$ for $\frac{\partial N}{\partial x}$ .

$4 y^{3} x^{2} = 3 y^{3} x^{2}$

Step 3.2

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

$4 y^{3} x^{2} = 3 y^{3} x^{2}$ 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 $4 y^{3} x^{2}$ for $\frac{\partial M}{\partial y}$ .

$\frac{4 y^{3} x^{2} - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $3 y^{3} x^{2}$ for $\frac{\partial N}{\partial x}$ .

$\frac{4 y^{3} x^{2} - (3 y^{3} x^{2})}{N}$

Step 4.3

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

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Step 4.3.1

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

$\frac{4 y^{3} x^{2} - (3 y^{3} x^{2})}{x^{3} y^{3}}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $y^{3} x^{2}$ out of $4 y^{3} x^{2} - 1 \cdot 3 y^{3} x^{2}$ .

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Step 4.3.2.1.1

Factor $y^{3} x^{2}$ out of $4 y^{3} x^{2}$ .

$\frac{y^{3} x^{2} (4) - 1 \cdot 3 y^{3} x^{2}}{x^{3} y^{3}}$

Step 4.3.2.1.2

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

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

Step 4.3.2.1.3

Factor $y^{3} x^{2}$ out of $y^{3} x^{2} (4) + y^{3} x^{2} (- 1 \cdot 3)$ .

$\frac{y^{3} x^{2} (4 - 1 \cdot 3)}{x^{3} y^{3}}$

Step 4.3.2.2

Multiply $- 1$ by $3$ .

$\frac{y^{3} x^{2} (4 - 3)}{x^{3} y^{3}}$

Step 4.3.2.3

Subtract $3$ from $4$ .

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

Step 4.3.2.4

Multiply $y^{3}$ by $1$ .

$\frac{x^{2} \cdot y^{3}}{x^{3} y^{3}}$

Step 4.3.3

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

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Step 4.3.3.1

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

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

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

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

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

Step 4.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.2.3

Rewrite the expression.

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

Step 4.3.4

Cancel the common factor of $y^{3}$ .

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Step 4.3.4.1

Cancel the common factor.

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

Step 4.3.4.2

Rewrite the expression.

$\frac{1}{x}$

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

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

$μ (x, y) = e^{\ln (| x |) + C}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Simplify.

$μ (x, y) = e^{\ln (| x |)} + C$

Step 5.2.2

Exponentiation and log are inverse functions.

$μ (x, y) = | x | + C$

Step 6

Multiply both sides of $(x^{2} y^{4} - 1) d x + x^{3} y^{3} d y = 0$ by the integration factor $x$ .

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Step 6.1

Multiply $x^{2} y^{4} - 1$ by $x$ .

$(x^{2} y^{4} - 1) x d x + x^{3} y^{3} d y = 0$

Step 6.2

Apply the distributive property.

$(x^{2} y^{4} x - 1 x) d x + x^{3} y^{3} d y = 0$

Step 6.3

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

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Step 6.3.1

Move $x$ .

$(x \cdot x^{2} y^{4} - 1 x) d x + x^{3} y^{3} d y = 0$

Step 6.3.2

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

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Step 6.3.2.1

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

$(x^{1} x^{2} y^{4} - 1 x) d x + x^{3} y^{3} d y = 0$

Step 6.3.2.2

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

$(x^{1 + 2} y^{4} - 1 x) d x + x^{3} y^{3} d y = 0$

Step 6.3.3

Add $1$ and $2$ .

$(x^{3} y^{4} - 1 x) d x + x^{3} y^{3} d y = 0$

Step 6.4

Rewrite $- 1 x$ as $- x$ .

$(x^{3} y^{4} - x) d x + x^{3} y^{3} d y = 0$

Step 6.5

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

$(x^{3} y^{4} - x) d x + x^{3} y^{3} x d y = 0$

Step 6.6

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

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Step 6.6.1

Move $x$ .

$(x^{3} y^{4} - x) d x + x \cdot x^{3} y^{3} d y = 0$

Step 6.6.2

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

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Step 6.6.2.1

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

$(x^{3} y^{4} - x) d x + x^{1} x^{3} y^{3} d y = 0$

Step 6.6.2.2

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

$(x^{3} y^{4} - x) d x + x^{1 + 3} y^{3} d y = 0$

Step 6.6.3

Add $1$ and $3$ .

$(x^{3} y^{4} - x) d x + x^{4} y^{3} d y = 0$

Step 7

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

$f (x, y) = \int x^{4} y^{3} d y$

Step 8

Integrate $N (x, y) = x^{4} y^{3}$ to find $f (x, y)$ .

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Step 8.1

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

$f (x, y) = x^{4} \int y^{3} d y$

Step 8.2

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

$f (x, y) = x^{4} (\frac{1}{4} y^{4} + C)$

Step 8.3

Simplify the answer.

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Step 8.3.1

Rewrite $x^{4} (\frac{1}{4} y^{4} + C)$ as $x^{4} \frac{1}{4} y^{4} + C$ .

$f (x, y) = x^{4} \frac{1}{4} y^{4} + C$

Step 8.3.2

Simplify.

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Step 8.3.2.1

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

$f (x, y) = \frac{x^{4}}{4} y^{4} + C$

Step 8.3.2.2

Combine $\frac{x^{4}}{4}$ and $y^{4}$ .

$f (x, y) = \frac{x^{4} y^{4}}{4} + C$

Step 8.3.3

Reorder terms.

$f (x, y) = \frac{1}{4} x^{4} y^{4} + C$

Step 9

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

$f (x, y) = \frac{1}{4} x^{4} y^{4} + g (x)$

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

By the Sum Rule, the derivative of $\frac{1}{4} x^{4} y^{4} + g (x)$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} x^{4} y^{4}] + \frac{d}{d x} [g (x)]$ .

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

Step 11.3

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{4} y^{4}]$ .

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Step 11.3.1

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

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

Step 11.3.2

Combine $\frac{x^{4}}{4}$ and $y^{4}$ .

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

Step 11.3.3

Since $\frac{y^{4}}{4}$ is constant with respect to $x$ , the derivative of $\frac{x^{4} y^{4}}{4}$ with respect to $x$ is $\frac{y^{4}}{4} \frac{d}{d x} [x^{4}]$ .

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

Step 11.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 11.3.5

Combine $4$ and $\frac{y^{4}}{4}$ .

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

Step 11.3.6

Combine $\frac{4 y^{4}}{4}$ and $x^{3}$ .

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

Step 11.3.7

Cancel the common factor of $4$ .

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Step 11.3.7.1

Cancel the common factor.

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

Step 11.3.7.2

Divide $y^{4} x^{3}$ by $1$ .

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

Subtract $y^{4} x^{3}$ from both sides of the equation.

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

Step 12.1.2

Combine the opposite terms in $x^{3} y^{4} - x - y^{4} x^{3}$ .

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Step 12.1.2.1

Reorder the factors in the terms $x^{3} y^{4}$ and $- y^{4} x^{3}$ .

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

Step 12.1.2.2

Subtract $y^{4} x^{3}$ from $y^{4} x^{3}$ .

$\frac{d g}{d x} = - x + 0$

Step 12.1.2.3

Add $- x$ and $0$ .

$\frac{d g}{d x} = - x$

Step 13

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

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Step 13.1

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

$\int \frac{d g}{d x} d x = \int - x d x$

Step 13.2

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

$g (x) = \int - x d x$

Step 13.3

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

$g (x) = - \int x d x$

Step 13.4

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

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

Step 13.5

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

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

Step 14

Substitute for $g (x)$ in $f (x, y) = \frac{1}{4} x^{4} y^{4} + g (x)$ .

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

Step 15

Simplify each term.

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Step 15.1

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

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

Step 15.2

Combine $\frac{x^{4}}{4}$ and $y^{4}$ .

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

Step 15.3

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

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