Calculus Examples

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

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = y (x + y + 1)$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [y (x + y + 1)]$

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y$ and $g (y) = x + y + 1$ .

$\frac{\partial M}{\partial y} = y \frac{d}{d y} [x + y + 1] + (x + y + 1) \frac{d}{d y} [y]$

Step 1.3

Differentiate.

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = y (\frac{d}{d y} [x] + \frac{d}{d y} [y] + \frac{d}{d y} [1]) + (x + y + 1) \frac{d}{d y} [y]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = y (0 + \frac{d}{d y} [y] + \frac{d}{d y} [1]) + (x + y + 1) \frac{d}{d y} [y]$

Step 1.3.3

Add $0$ and $\frac{d}{d y} [y]$ .

$\frac{\partial M}{\partial y} = y (\frac{d}{d y} [y] + \frac{d}{d y} [1]) + (x + y + 1) \frac{d}{d y} [y]$

Step 1.3.4

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

$\frac{\partial M}{\partial y} = y (1 + \frac{d}{d y} [1]) + (x + y + 1) \frac{d}{d y} [y]$

Step 1.3.5

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

$\frac{\partial M}{\partial y} = y (1 + 0) + (x + y + 1) \frac{d}{d y} [y]$

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add $1$ and $0$ .

$\frac{\partial M}{\partial y} = y \cdot 1 + (x + y + 1) \frac{d}{d y} [y]$

Step 1.3.6.2

Multiply $y$ by $1$ .

$\frac{\partial M}{\partial y} = y + (x + y + 1) \frac{d}{d y} [y]$

Step 1.3.7

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

$\frac{\partial M}{\partial y} = y + (x + y + 1) \cdot 1$

Step 1.3.8

Simplify by adding terms.

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Step 1.3.8.1

Multiply $x + y + 1$ by $1$ .

$\frac{\partial M}{\partial y} = y + x + y + 1$

Step 1.3.8.2

Add $y$ and $y$ .

$\frac{\partial M}{\partial y} = x + 2 y + 1$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x + 3 y + 2$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

$\frac{\partial N}{\partial x} = x (1 + 0 + \frac{d}{d x} [2]) + (x + 3 y + 2) \frac{d}{d x} [x]$

Step 2.3.4

Add $1$ and $0$ .

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

Step 2.3.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = x (1 + 0) + (x + 3 y + 2) \frac{d}{d x} [x]$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $1$ and $0$ .

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

Step 2.3.6.2

Multiply $x$ by $1$ .

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

Step 2.3.7

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

$\frac{\partial N}{\partial x} = x + (x + 3 y + 2) \cdot 1$

Step 2.3.8

Simplify by adding terms.

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Step 2.3.8.1

Multiply $x + 3 y + 2$ by $1$ .

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

Step 2.3.8.2

Add $x$ and $x$ .

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

Step 3

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

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Step 3.1

Substitute $x + 2 y + 1$ for $\frac{\partial M}{\partial y}$ and $2 x + 3 y + 2$ for $\frac{\partial N}{\partial x}$ .

$x + 2 y + 1 = 2 x + 3 y + 2$

Step 3.2

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

$x + 2 y + 1 = 2 x + 3 y + 2$ is not an identity.

Step 4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} d y}$ .

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Step 4.1

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

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

Step 4.2

Substitute $x + 2 y + 1$ for $\frac{\partial M}{\partial y}$ .

$\frac{2 x + 3 y + 2 - (x + 2 y + 1)}{M}$

Step 4.3

Substitute $y (x + y + 1)$ for $M$ .

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Step 4.3.1

Substitute $y (x + y + 1)$ for $M$ .

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

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

Step 4.3.2.2

Simplify.

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Step 4.3.2.2.1

Multiply $2$ by $- 1$ .

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

Step 4.3.2.2.2

Multiply $- 1$ by $1$ .

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

Step 4.3.2.3

Subtract $x$ from $2 x$ .

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

Step 4.3.2.4

Subtract $2 y$ from $3 y$ .

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

Step 4.3.2.5

Subtract $1$ from $2$ .

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

Step 4.3.3

Substitute $y (x + y + 1)$ for $M$ .

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Step 4.3.3.1

Cancel the common factor.

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

Step 4.3.3.2

Rewrite the expression.

$\frac{1}{y}$

Step 4.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} d y}$ .

$μ (x, y) = e^{\int \frac{1}{y} d y}$

Step 5

Evaluate the integral $e^{\int \frac{1}{y} d y}$ .

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Step 5.1

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

$μ (x, y) = e^{\ln (| y |) + C}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Simplify.

$μ (x, y) = e^{\ln (| y |)} + C$

Step 5.2.2

Exponentiation and log are inverse functions.

$μ (x, y) = | y | + C$

Step 6

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

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Step 6.1

Multiply $y (x + y + 1)$ by $y$ .

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

Step 6.2

Multiply $y$ by $y$ by adding the exponents.

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Step 6.2.1

Move $y$ .

$y \cdot y (x + y + 1) d x + x (x + 3 y + 2) d y = 0$

Step 6.2.2

Multiply $y$ by $y$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Simplify.

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Step 6.4.1

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

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Step 6.4.1.1

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

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Step 6.4.1.1.1

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

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

Step 6.4.1.1.2

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

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

Step 6.4.1.2

Add $2$ and $1$ .

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

Step 6.4.2

Multiply $y^{2}$ by $1$ .

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

Step 6.5

Multiply $x (x + 3 y + 2)$ by $y$ .

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

Step 6.6

Apply the distributive property.

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

Step 6.7

Simplify.

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Step 6.7.1

Multiply $x$ by $x$ .

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

Step 6.7.2

Rewrite using the commutative property of multiplication.

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

Step 6.7.3

Move $2$ to the left of $x$ .

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

Step 6.8

Apply the distributive property.

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

Step 6.9

Multiply $y$ by $y$ by adding the exponents.

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Step 6.9.1

Move $y$ .

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

Step 6.9.2

Multiply $y$ by $y$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

Split the single integral into multiple integrals.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Apply the constant rule.

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

Step 8.5

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

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

Step 8.6

Apply the constant rule.

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

Step 8.7

Simplify.

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

Step 8.8

Reorder terms.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.3.4

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

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

Step 11.3.5

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

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

Step 11.3.6

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

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

Step 11.3.7

Cancel the common factor of $2$ .

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Step 11.3.7.1

Cancel the common factor.

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

Step 11.3.7.2

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

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

Step 11.4

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

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Step 11.4.1

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

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

Step 11.4.2

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

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

Step 11.4.3

Move $3$ to the left of $x$ .

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

Step 11.5

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

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Step 11.5.1

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

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

Step 11.5.2

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

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

Step 11.5.3

Move $2$ to the left of $x$ .

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

Step 11.6

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

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

Step 11.7

Reorder terms.

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

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

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

Step 12.1.2

Subtract $3 y^{2} x$ from both sides of the equation.

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

Step 12.1.3

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

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

Step 12.1.4

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

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Step 12.1.4.1

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

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

Step 12.1.4.2

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

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

Step 12.1.4.3

Reorder the factors in the terms $3 x y^{2}$ and $- 3 y^{2} x$ .

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

Step 12.1.4.4

Subtract $3 y^{2} x$ from $3 y^{2} x$ .

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

Step 12.1.4.5

Add $2 x y$ and $0$ .

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

Step 12.1.4.6

Subtract $2 x y$ from $2 x y$ .

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

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

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

Step 15

Simplify each term.

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Step 15.1

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

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

Step 15.2

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

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