Calculus Examples

$x (x + 2 y) d y + (4 x y + 3 y^{2} - x) d x = 0$

Step 1

Rewrite the differential equation to fit the Exact differential equation technique.

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Step 1.1

Rewrite.

$(4 x y + 3 y^{2} - x) d x + x (x + 2 y) d y = 0$

Step 2

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 4 x y + 3 y^{2} - x$ .

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Step 2.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [4 x y + 3 y^{2} - x]$

Step 2.2

By the Sum Rule, the derivative of $4 x y + 3 y^{2} - x$ with respect to $y$ is $\frac{d}{d y} [4 x y] + \frac{d}{d y} [3 y^{2}] + \frac{d}{d y} [- x]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [4 x y] + \frac{d}{d y} [3 y^{2}] + \frac{d}{d y} [- x]$

Step 2.3

Evaluate $\frac{d}{d y} [4 x y]$ .

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Step 2.3.1

Since $4 x$ is constant with respect to $y$ , the derivative of $4 x y$ with respect to $y$ is $4 x \frac{d}{d y} [y]$ .

$\frac{\partial M}{\partial y} = 4 x \frac{d}{d y} [y] + \frac{d}{d y} [3 y^{2}] + \frac{d}{d y} [- x]$

Step 2.3.2

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

$\frac{\partial M}{\partial y} = 4 x \cdot 1 + \frac{d}{d y} [3 y^{2}] + \frac{d}{d y} [- x]$

Step 2.3.3

Multiply $4$ by $1$ .

$\frac{\partial M}{\partial y} = 4 x + \frac{d}{d y} [3 y^{2}] + \frac{d}{d y} [- x]$

Step 2.4

Evaluate $\frac{d}{d y} [3 y^{2}]$ .

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Step 2.4.1

Since $3$ is constant with respect to $y$ , the derivative of $3 y^{2}$ with respect to $y$ is $3 \frac{d}{d y} [y^{2}]$ .

$\frac{\partial M}{\partial y} = 4 x + 3 \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- x]$

Step 2.4.2

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

$\frac{\partial M}{\partial y} = 4 x + 3 (2 y) + \frac{d}{d y} [- x]$

Step 2.4.3

Multiply $2$ by $3$ .

$\frac{\partial M}{\partial y} = 4 x + 6 y + \frac{d}{d y} [- x]$

Step 2.5

Differentiate using the Constant Rule.

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Step 2.5.1

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

$\frac{\partial M}{\partial y} = 4 x + 6 y + 0$

Step 2.5.2

Add $4 x + 6 y$ and $0$ .

$\frac{\partial M}{\partial y} = 4 x + 6 y$

Step 3

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

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Step 3.1

Differentiate $N$ with respect to $x$ .

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

Step 3.2

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 $g (x) = x + 2 y$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Add $1$ and $0$ .

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

Step 3.3.4.2

Multiply $x$ by $1$ .

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

Step 3.3.5

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

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

Step 3.3.6

Simplify by adding terms.

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Step 3.3.6.1

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

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

Step 3.3.6.2

Add $x$ and $x$ .

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

Step 4

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

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Step 4.1

Substitute $4 x + 6 y$ for $\frac{\partial M}{\partial y}$ and $2 x + 2 y$ for $\frac{\partial N}{\partial x}$ .

$4 x + 6 y = 2 x + 2 y$

Step 4.2

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

$4 x + 6 y = 2 x + 2 y$ is not an identity.

Step 5

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 5.1

Substitute $4 x + 6 y$ for $\frac{\partial M}{\partial y}$ .

$\frac{4 x + 6 y - \frac{\partial N}{\partial x}}{N}$

Step 5.2

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

$\frac{4 x + 6 y - (2 x + 2 y)}{N}$

Step 5.3

Substitute $x (x + 2 y)$ for $N$ .

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Step 5.3.1

Substitute $x (x + 2 y)$ for $N$ .

$\frac{4 x + 6 y - (2 x + 2 y)}{x (x + 2 y)}$

Step 5.3.2

Simplify the numerator.

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Step 5.3.2.1

Apply the distributive property.

$\frac{4 x + 6 y - (2 x) - (2 y)}{x (x + 2 y)}$

Step 5.3.2.2

Multiply $2$ by $- 1$ .

$\frac{4 x + 6 y - 2 x - (2 y)}{x (x + 2 y)}$

Step 5.3.2.3

Multiply $2$ by $- 1$ .

$\frac{4 x + 6 y - 2 x - 2 y}{x (x + 2 y)}$

Step 5.3.2.4

Subtract $2 x$ from $4 x$ .

$\frac{2 x + 6 y - 2 y}{x (x + 2 y)}$

Step 5.3.2.5

Subtract $2 y$ from $6 y$ .

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

Step 5.3.2.6

Factor $2$ out of $2 x + 4 y$ .

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Step 5.3.2.6.1

Factor $2$ out of $2 x$ .

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

Step 5.3.2.6.2

Factor $2$ out of $4 y$ .

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

Step 5.3.2.6.3

Factor $2$ out of $2 (x) + 2 (2 y)$ .

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

Step 5.3.3

Cancel the common factor of $x + 2 y$ .

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Step 5.3.3.1

Cancel the common factor.

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

Step 5.3.3.2

Rewrite the expression.

$\frac{2}{x}$

Step 5.4

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

$μ (x, y) = e^{\int \frac{2}{x} d x}$

Step 6

Evaluate the integral $e^{\int \frac{2}{x} d x}$ .

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Step 6.1

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

$μ (x, y) = e^{2 \int \frac{1}{x} d x}$

Step 6.2

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

$μ (x, y) = e^{2 (\ln (| x |) + C)}$

Step 6.3

Simplify.

$μ (x, y) = e^{2 \ln (| x |)} + C$

Step 6.4

Simplify each term.

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Step 6.4.1

Simplify $2 \ln (| x |)$ by moving $2$ inside the logarithm.

$μ (x, y) = e^{\ln ({| x |}^{2})} + C$

Step 6.4.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| x |}^{2} + C$

Step 6.4.3

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$μ (x, y) = x^{2} + C$

Step 7

Multiply both sides of $(4 x y + 3 y^{2} - x) d x + x (x + 2 y) d y = 0$ by the integration factor $x^{2}$ .

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Step 7.1

Multiply $4 x y + 3 y^{2} - x$ by $x^{2}$ .

$(4 x y + 3 y^{2} - x) x^{2} d x + x (x + 2 y) d y = 0$

Step 7.2

Apply the distributive property.

$(4 x y x^{2} + 3 y^{2} x^{2} - x \cdot x^{2}) d x + x (x + 2 y) d y = 0$

Step 7.3

Simplify.

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Step 7.3.1

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

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Step 7.3.1.1

Move $x^{2}$ .

$(4 (x^{2} x) y + 3 y^{2} x^{2} - x \cdot x^{2}) d x + x (x + 2 y) d y = 0$

Step 7.3.1.2

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

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Step 7.3.1.2.1

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

$(4 (x^{2} x^{1}) y + 3 y^{2} x^{2} - x \cdot x^{2}) d x + x (x + 2 y) d y = 0$

Step 7.3.1.2.2

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

$(4 x^{2 + 1} y + 3 y^{2} x^{2} - x \cdot x^{2}) d x + x (x + 2 y) d y = 0$

Step 7.3.1.3

Add $2$ and $1$ .

$(4 x^{3} y + 3 y^{2} x^{2} - x \cdot x^{2}) d x + x (x + 2 y) d y = 0$

Step 7.3.2

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

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Step 7.3.2.1

Move $x^{2}$ .

$(4 x^{3} y + 3 y^{2} x^{2} - (x^{2} x)) d x + x (x + 2 y) d y = 0$

Step 7.3.2.2

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

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Step 7.3.2.2.1

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

$(4 x^{3} y + 3 y^{2} x^{2} - (x^{2} x^{1})) d x + x (x + 2 y) d y = 0$

Step 7.3.2.2.2

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

$(4 x^{3} y + 3 y^{2} x^{2} - x^{2 + 1}) d x + x (x + 2 y) d y = 0$

Step 7.3.2.3

Add $2$ and $1$ .

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + x (x + 2 y) d y = 0$

Step 7.4

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

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + x (x + 2 y) x^{2} d y = 0$

Step 7.5

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

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Step 7.5.1

Move $x^{2}$ .

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + x^{2} x (x + 2 y) d y = 0$

Step 7.5.2

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

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Step 7.5.2.1

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

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + x^{2} x^{1} (x + 2 y) d y = 0$

Step 7.5.2.2

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

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + x^{2 + 1} (x + 2 y) d y = 0$

Step 7.5.3

Add $2$ and $1$ .

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + x^{3} (x + 2 y) d y = 0$

Step 7.6

Apply the distributive property.

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + (x^{3} x + x^{3} (2 y)) d y = 0$

Step 7.7

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 7.7.1

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

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Step 7.7.1.1

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

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + (x^{3} x^{1} + x^{3} (2 y)) d y = 0$

Step 7.7.1.2

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

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + (x^{3 + 1} + x^{3} (2 y)) d y = 0$

Step 7.7.2

Add $3$ and $1$ .

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + (x^{4} + x^{3} (2 y)) d y = 0$

Step 7.8

Rewrite using the commutative property of multiplication.

$(4 x^{3} y + 3 y^{2} x^{2} - x^{3}) d x + (x^{4} + 2 x^{3} y) d y = 0$

Step 8

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

$f (x, y) = \int x^{4} + 2 x^{3} y d y$

Step 9

Integrate $N (x, y) = x^{4} + 2 x^{3} y$ to find $f (x, y)$ .

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Step 9.1

Split the single integral into multiple integrals.

$f (x, y) = \int x^{4} d y + \int 2 x^{3} y d y$

Step 9.2

Apply the constant rule.

$f (x, y) = x^{4} y + C + \int 2 x^{3} y d y$

Step 9.3

Since $2 x^{3}$ is constant with respect to $y$ , move $2 x^{3}$ out of the integral.

$f (x, y) = x^{4} y + C + 2 x^{3} \int y d y$

Step 9.4

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

$f (x, y) = x^{4} y + C + 2 x^{3} (\frac{1}{2} y^{2} + C)$

Step 9.5

Simplify.

$f (x, y) = x^{4} y + 2 \cdot \frac{1}{2} x^{3} y^{2} + C$

Step 9.6

Simplify.

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Step 9.6.1

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

$f (x, y) = x^{4} y + \frac{2}{2} x^{3} y^{2} + C$

Step 9.6.2

Cancel the common factor of $2$ .

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Step 9.6.2.1

Cancel the common factor.

$f (x, y) = x^{4} y + \frac{2}{2} x^{3} y^{2} + C$

Step 9.6.2.2

Rewrite the expression.

$f (x, y) = x^{4} y + 1 x^{3} y^{2} + C$

Step 9.6.3

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

$f (x, y) = x^{4} y + x^{3} y^{2} + C$

Step 10

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

$f (x, y) = x^{4} y + x^{3} y^{2} + g (x)$

Step 11

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

$\frac{\partial f}{\partial x} = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 12

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

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Step 12.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [x^{4} y + x^{3} y^{2} + g (x)] = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 12.2

By the Sum Rule, the derivative of $x^{4} y + x^{3} y^{2} + g (x)$ with respect to $x$ is $\frac{d}{d x} [x^{4} y] + \frac{d}{d x} [x^{3} y^{2}] + \frac{d}{d x} [g (x)]$ .

$\frac{d}{d x} [x^{4} y] + \frac{d}{d x} [x^{3} y^{2}] + \frac{d}{d x} [g (x)] = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 12.3

Evaluate $\frac{d}{d x} [x^{4} y]$ .

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Step 12.3.1

Since $y$ is constant with respect to $x$ , the derivative of $x^{4} y$ with respect to $x$ is $y \frac{d}{d x} [x^{4}]$ .

$y \frac{d}{d x} [x^{4}] + \frac{d}{d x} [x^{3} y^{2}] + \frac{d}{d x} [g (x)] = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 12.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$y (4 x^{3}) + \frac{d}{d x} [x^{3} y^{2}] + \frac{d}{d x} [g (x)] = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 12.3.3

Move $4$ to the left of $y$ .

$4 y x^{3} + \frac{d}{d x} [x^{3} y^{2}] + \frac{d}{d x} [g (x)] = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 12.4

Evaluate $\frac{d}{d x} [x^{3} y^{2}]$ .

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Step 12.4.1

Since $y^{2}$ is constant with respect to $x$ , the derivative of $x^{3} y^{2}$ with respect to $x$ is $y^{2} \frac{d}{d x} [x^{3}]$ .

$4 y x^{3} + y^{2} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [g (x)] = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 12.4.2

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

$4 y x^{3} + y^{2} (3 x^{2}) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 12.4.3

Move $3$ to the left of $y^{2}$ .

$4 y x^{3} + 3 y^{2} x^{2} + \frac{d}{d x} [g (x)] = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 12.5

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

$4 y x^{3} + 3 y^{2} x^{2} + \frac{d g}{d x} = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 12.6

Reorder terms.

$\frac{d g}{d x} + 4 x^{3} y + 3 x^{2} y^{2} = 4 x^{3} y + 3 y^{2} x^{2} - x^{3}$

Step 13

Solve for $\frac{d g}{d x}$ .

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Step 13.1

Move all terms not containing $\frac{d g}{d x}$ to the right side of the equation.

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Step 13.1.1

Subtract $4 x^{3} y$ from both sides of the equation.

$\frac{d g}{d x} + 3 x^{2} y^{2} = 4 x^{3} y + 3 y^{2} x^{2} - x^{3} - 4 x^{3} y$

Step 13.1.2

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

$\frac{d g}{d x} = 4 x^{3} y + 3 y^{2} x^{2} - x^{3} - 4 x^{3} y - 3 x^{2} y^{2}$

Step 13.1.3

Combine the opposite terms in $4 x^{3} y + 3 y^{2} x^{2} - x^{3} - 4 x^{3} y - 3 x^{2} y^{2}$ .

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Step 13.1.3.1

Subtract $4 x^{3} y$ from $4 x^{3} y$ .

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

Step 13.1.3.2

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

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

Step 13.1.3.3

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

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

Step 13.1.3.4

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

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

Step 13.1.3.5

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

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

Step 14

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

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Step 14.1

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 14.2

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

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

Step 14.3

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

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

Step 14.4

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 14.5

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

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

Step 15

Substitute for $g (x)$ in $f (x, y) = x^{4} y + x^{3} y^{2} + g (x)$ .

$f (x, y) = x^{4} y + x^{3} y^{2} - \frac{1}{4} x^{4} + C$

Step 16

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

$f (x, y) = x^{4} y + x^{3} y^{2} - \frac{x^{4}}{4} + C$