Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $3$ by $1$ .

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

Step 1.4

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

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$\frac{\partial N}{\partial x} = y \cdot 1 + \frac{d}{d x} [x^{2}]$

Step 2.3.3

Multiply $y$ by $1$ .

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

Step 2.4

Differentiate using the Power Rule.

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Step 2.4.1

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

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

Step 2.4.2

Reorder terms.

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

Step 3

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

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Step 3.1

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

$3 x + 2 y = 2 x + y$

Step 3.2

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

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

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

Step 4.2

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

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

Step 4.3

Substitute $x y + x^{2}$ for $N$ .

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Step 4.3.1

Substitute $x y + x^{2}$ for $N$ .

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

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

Step 4.3.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.2.3

Subtract $2 x$ from $3 x$ .

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

Step 4.3.2.4

Subtract $y$ from $2 y$ .

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

Step 4.3.3

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

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Step 4.3.3.1

Factor $x$ out of $x y$ .

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

Step 4.3.3.2

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

$\frac{x + y}{x (y) + x \cdot x}$

Step 4.3.3.3

Factor $x$ out of $x (y) + x \cdot x$ .

$\frac{x + y}{x (y + x)}$

Step 4.3.4

Cancel the common factor of $x + y$ and $y + x$ .

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Step 4.3.4.1

Reorder terms.

$\frac{x + y}{x (x + y)}$

Step 4.3.4.2

Cancel the common factor.

$\frac{x + y}{x (x + y)}$

Step 4.3.4.3

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

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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Step 6.3.1

Move $x$ .

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

Step 6.3.2

Multiply $x$ by $x$ .

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

Step 6.4

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

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

Step 6.5

Apply the distributive property.

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

Step 6.6

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

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Step 6.6.1

Move $x$ .

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

Step 6.6.2

Multiply $x$ by $x$ .

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

Step 6.7

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

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Step 6.7.1

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

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Step 6.7.1.1

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

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

Step 6.7.1.2

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

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

Step 6.7.2

Add $2$ and $1$ .

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

Step 7

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

$f (x, y) = \int 3 x^{2} y + y^{2} x d x$

Step 8

Integrate $M (x, y) = 3 x^{2} y + y^{2} x$ to find $f (x, y)$ .

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Step 8.1

Split the single integral into multiple integrals.

$f (x, y) = \int 3 x^{2} y d x + \int y^{2} x d x$

Step 8.2

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

$f (x, y) = 3 y \int x^{2} d x + \int y^{2} x d x$

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

Simplify.

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

Step 8.7

Simplify.

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Step 8.7.1

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

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

Step 8.7.2

Cancel the common factor of $3$ .

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Step 8.7.2.1

Cancel the common factor.

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

Step 8.7.2.2

Rewrite the expression.

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

Step 8.7.3

Multiply $y$ by $1$ .

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

Step 8.7.4

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

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

Step 8.7.5

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

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

Step 8.8

Reorder terms.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

By the Sum Rule, the derivative of $y x^{3} + \frac{1}{2} y^{2} x^{2} + g (y)$ with respect to $y$ is $\frac{d}{d y} [y x^{3}] + \frac{d}{d y} [\frac{1}{2} y^{2} x^{2}] + \frac{d}{d y} [g (y)]$ .

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.4

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

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Step 11.4.1

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

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

Step 11.4.2

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

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

Step 11.4.3

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

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

Step 11.4.4

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

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

Step 11.4.5

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

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

Step 11.4.6

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

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

Step 11.4.7

Cancel the common factor of $2$ .

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Step 11.4.7.1

Cancel the common factor.

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

Step 11.4.7.2

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

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

Step 11.5

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

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

Step 11.6

Reorder terms.

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

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

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

Step 12.1.2

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

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

Step 12.1.3

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

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Step 12.1.3.1

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

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

Step 12.1.3.2

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

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

Step 12.1.3.3

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

$\frac{d g}{d y} = 0$

Step 13

Find the antiderivative of $0$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = 0$ .

$\int \frac{d g}{d y} d y = \int 0 d y$

Step 13.2

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

$g (y) = \int 0 d y$

Step 13.3

The integral of $0$ with respect to $y$ is $0$ .

$g (y) = 0 + C$

Step 13.4

Add $0$ and $C$ .

$g (y) = C$

Step 14

Substitute for $g (y)$ in $f (x, y) = y x^{3} + \frac{1}{2} y^{2} x^{2} + g (y)$ .

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

Step 15

Simplify each term.

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Step 15.1

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

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

Step 15.2

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

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