Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Add $2 y$ and $0$ .

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

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 $y$ 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

Subtract $y$ from $2 y$ .

$\frac{y}{x y}$

Step 4.3.3

Cancel the common factor of $y$ .

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Step 4.3.3.1

Cancel the common factor.

$\frac{y}{x y}$

Step 4.3.3.2

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

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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Step 6.3.1

Move $x$ .

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

Step 6.3.2

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

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Step 6.3.2.1

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

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

Step 6.3.2.2

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

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

Step 6.3.3

Add $1$ and $2$ .

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

Step 6.4

Multiply $x y$ by $x$ .

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

Step 6.5

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

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Step 6.5.1

Move $x$ .

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

Step 6.5.2

Multiply $x$ by $x$ .

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

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

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

Step 8.3.2.2

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

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

Step 8.3.3

Reorder terms.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

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

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

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

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

Step 11.3.4

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

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

Step 11.3.5

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

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

Step 11.3.6

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

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

Step 11.3.7

Cancel the common factor of $2$ .

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Step 11.3.7.1

Cancel the common factor.

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

Step 11.3.7.2

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

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

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

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

Step 12.1.2

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

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Step 12.1.2.1

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

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

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

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

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

Step 13.4

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 13.5

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

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

Step 14

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

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

Step 15

Simplify each term.

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Step 15.1

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

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

Step 15.2

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

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

Step 15.3

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

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