Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $2$ .

$\frac{\partial M}{\partial y} = 0 + 4 y$

Step 1.4

Add $0$ and $4 y$ .

$\frac{\partial M}{\partial y} = 4 y$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $1$ .

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

Step 3

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

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Step 3.1

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

$4 y = 2 y$

Step 3.2

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

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

$\frac{4 y - \frac{\partial N}{\partial x}}{N}$

Step 4.2

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

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

Step 4.3

Substitute $2 x y$ for $N$ .

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Step 4.3.1

Substitute $2 x y$ for $N$ .

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $y$ out of $4 y - 1 \cdot 2 y$ .

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Step 4.3.2.1.1

Factor $y$ out of $4 y$ .

$\frac{y \cdot 4 - 1 \cdot 2 y}{2 x y}$

Step 4.3.2.1.2

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

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

Step 4.3.2.1.3

Factor $y$ out of $y \cdot 4 + y (- 1 \cdot 2)$ .

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

Step 4.3.2.2

Multiply $- 1$ by $2$ .

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

Step 4.3.2.3

Subtract $2$ from $4$ .

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

Step 4.3.3

Cancel the common factor of $y$ .

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Step 4.3.3.1

Cancel the common factor.

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

Step 4.3.3.2

Rewrite the expression.

$\frac{2}{2 x}$

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Cancel the common factor.

$\frac{2}{2 x}$

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 $(3 x + 2 y^{2}) d x + 2 x y d y = 0$ by the integration factor $x$ .

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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Step 6.3.1

Move $x$ .

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

Step 6.3.2

Multiply $x$ by $x$ .

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

Step 6.4

Multiply $2 x y$ by $x$ .

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

Step 6.5

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

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Step 6.5.1

Move $x$ .

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

Step 6.5.2

Multiply $x$ by $x$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Simplify the answer.

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Step 8.3.1

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

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

Step 8.3.2

Simplify.

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Step 8.3.2.1

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

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

Step 8.3.2.2

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

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

Step 8.3.2.3

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

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

Step 8.3.2.4

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

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

Step 8.3.2.5

Cancel the common factor of $2$ .

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Step 8.3.2.5.1

Cancel the common factor.

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

Step 8.3.2.5.2

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

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

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

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

Step 12.1.2

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

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Step 12.1.2.1

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

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

Step 12.1.2.2

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

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

Step 13

Find the antiderivative of $3 x^{2}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = 3 x^{2}$ .

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

Step 13.2

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

$g (x) = \int 3 x^{2} d x$

Step 13.3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$g (x) = 3 \int x^{2} d x$

Step 13.4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$g (x) = 3 (\frac{1}{3} x^{3} + C)$

Step 13.5

Simplify the answer.

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Step 13.5.1

Rewrite $3 (\frac{1}{3} x^{3} + C)$ as $3 (\frac{1}{3}) x^{3} + C$ .

$g (x) = 3 (\frac{1}{3}) x^{3} + C$

Step 13.5.2

Simplify.

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Step 13.5.2.1

Combine $3$ and $\frac{1}{3}$ .

$g (x) = \frac{3}{3} x^{3} + C$

Step 13.5.2.2

Cancel the common factor of $3$ .

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Step 13.5.2.2.1

Cancel the common factor.

$g (x) = \frac{3}{3} x^{3} + C$

Step 13.5.2.2.2

Rewrite the expression.

$g (x) = 1 x^{3} + C$

Step 13.5.2.3

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

$g (x) = x^{3} + C$

Step 14

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

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