Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 0 - \frac{d}{d y} [y^{2}]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 0 - (2 y)$

Step 1.3.3

Multiply $2$ by $- 1$ .

$\frac{\partial M}{\partial y} = 0 - 2 y$

Step 1.4

Subtract $2 y$ from $0$ .

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $- 1$ by $1$ .

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

Step 3

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

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Step 3.1

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

$- 2 y = - y$

Step 3.2

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

$- 2 y = - y$ 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 $- 2 y$ for $\frac{\partial M}{\partial y}$ .

$\frac{- 2 y - \frac{\partial N}{\partial x}}{N}$

Step 4.2

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

$\frac{- 2 y - - y}{N}$

Step 4.3

Substitute $- x y$ for $N$ .

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Step 4.3.1

Substitute $- x y$ for $N$ .

$\frac{- 2 y - - y}{- x y}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

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

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Step 4.3.2.1.1

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

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

Step 4.3.2.1.2

Factor $y$ out of $- - y$ .

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

Step 4.3.2.1.3

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

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

Step 4.3.2.2

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2.3

Add $- 2$ and $1$ .

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

Step 4.3.3

Cancel the common factor of $y$ .

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Step 4.3.3.1

Cancel the common factor.

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

Step 4.3.3.2

Rewrite the expression.

$\frac{- 1}{- x}$

Step 4.3.4

Dividing two negative values results in a positive value.

$\frac{1}{x}$

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 $(x^{2} - y^{2}) d x - x y d y = 0$ by the integration factor $x$ .

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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Step 6.3.1

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

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Step 6.3.1.1

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

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

Step 6.3.1.2

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

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

Step 6.3.2

Add $2$ and $1$ .

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

Step 6.4

Multiply $- x y$ by $x$ .

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

Step 6.5

Multiply $x$ by $x$ by adding the exponents.

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Step 6.5.1

Move $x$ .

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

Step 6.5.2

Multiply $x$ by $x$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Simplify the answer.

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Step 8.3.1

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

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

Step 8.3.2

Simplify.

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Step 8.3.2.1

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

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

Step 8.3.2.2

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

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

Step 8.3.3

Simplify.

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Step 8.3.3.1

Reorder terms.

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

Step 8.3.3.2

Remove parentheses.

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

Step 8.3.3.3

Remove parentheses.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

Evaluate $\frac{d}{d x} [- \frac{1}{2} y^{2} x^{2}]$ .

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.3.4

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

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

Step 11.3.5

Multiply $2$ by $- 1$ .

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

Step 11.3.6

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

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

Step 11.3.7

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

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

Step 11.3.8

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

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Step 11.3.8.1

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

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

Step 11.3.8.2

Cancel the common factors.

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Step 11.3.8.2.1

Factor $2$ out of $2$ .

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

Step 11.3.8.2.2

Cancel the common factor.

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

Step 11.3.8.2.3

Rewrite the expression.

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

Step 11.3.8.2.4

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

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

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

Add $y^{2} x$ to both sides of the equation.

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

Step 12.1.2

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

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Step 12.1.2.1

Add $- y^{2} x$ and $y^{2} x$ .

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

Step 12.1.2.2

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

$\frac{d g}{d x} = x^{3}$

Step 13

Find the antiderivative of $x^{3}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = x^{3}$ .

$\int \frac{d g}{d x} d x = \int x^{3} d x$

Step 13.2

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

$g (x) = \int x^{3} d x$

Step 13.3

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

$g (x) = \frac{1}{4} x^{4} + C$

Step 14

Substitute for $g (x)$ in $f (x, y) = - \frac{1}{2} y^{2} x^{2} + g (x)$ .

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

Step 15

Simplify each term.

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Step 15.1

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

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

Step 15.2

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

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

Step 15.3

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

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