Calculus Examples

$(y - x) d x + 4 x d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

$\frac{\partial M}{\partial y} = 1 + 0$

Step 1.5

Add $1$ and $0$ .

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

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = 4 x$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [4 x]$

Step 2.2

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

$\frac{\partial N}{\partial x} = 4 \frac{d}{d x} [x]$

Step 2.3

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

$\frac{\partial N}{\partial x} = 4 \cdot 1$

Step 2.4

Multiply $4$ by $1$ .

$\frac{\partial N}{\partial x} = 4$

Step 3

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

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Step 3.1

Substitute $1$ for $\frac{\partial M}{\partial y}$ and $4$ for $\frac{\partial N}{\partial x}$ .

$1 = 4$

Step 3.2

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

$1 = 4$ 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 $1$ for $\frac{\partial M}{\partial y}$ .

$\frac{1 - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $4$ for $\frac{\partial N}{\partial x}$ .

$\frac{1 - 4}{N}$

Step 4.3

Substitute $4 x$ for $N$ .

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Step 4.3.1

Substitute $4 x$ for $N$ .

$\frac{1 - 4}{4 x}$

Step 4.3.2

Subtract $4$ from $1$ .

$\frac{- 3}{4 x}$

Step 4.3.3

Move the negative in front of the fraction.

$- \frac{3}{4 x}$

Step 4.4

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

$μ (x, y) = e^{\int - \frac{3}{4 x} d x}$

Step 5

Evaluate the integral $e^{\int - \frac{3}{4 x} d x}$ .

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Step 5.1

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

$μ (x, y) = e^{- \int \frac{3}{4 x} d x}$

Step 5.2

Since $\frac{3}{4}$ is constant with respect to $x$ , move $\frac{3}{4}$ out of the integral.

$μ (x, y) = e^{- (\frac{3}{4} \int \frac{1}{x} d x)}$

Step 5.3

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

$μ (x, y) = e^{- \frac{3}{4} (\ln (| x |) + C)}$

Step 5.4

Simplify.

$μ (x, y) = e^{- \frac{3}{4} \ln (| x |)} + C$

Step 5.5

Simplify each term.

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Step 5.5.1

Multiply $- \frac{3}{4} \ln (| x |)$ .

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Step 5.5.1.1

Reorder $\ln (| x |)$ and $\frac{3}{4}$ .

$μ (x, y) = e^{- (\frac{3}{4} \ln (| x |))} + C$

Step 5.5.1.2

Simplify $\frac{3}{4} \ln (| x |)$ by moving $\frac{3}{4}$ inside the logarithm.

$μ (x, y) = e^{- \ln ({| x |}^{\frac{3}{4}})} + C$

Step 5.5.2

Simplify $- \ln ({| x |}^{\frac{3}{4}})$ by moving $- 1$ inside the logarithm.

$μ (x, y) = e^{\ln ({({| x |}^{\frac{3}{4}})}^{- 1})} + C$

Step 5.5.3

Exponentiation and log are inverse functions.

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

Step 5.5.4

Multiply the exponents in ${({| x |}^{\frac{3}{4}})}^{- 1}$ .

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Step 5.5.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$μ (x, y) = {| x |}^{\frac{3}{4} \cdot - 1} + C$

Step 5.5.4.2

Multiply $\frac{3}{4} \cdot - 1$ .

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Step 5.5.4.2.1

Combine $\frac{3}{4}$ and $- 1$ .

$μ (x, y) = {| x |}^{\frac{3 \cdot - 1}{4}} + C$

Step 5.5.4.2.2

Multiply $3$ by $- 1$ .

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

Step 5.5.4.3

Move the negative in front of the fraction.

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

Step 5.5.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 6

Multiply both sides of $(y - x) d x + 4 x d y = 0$ by the integration factor $\frac{1}{x^{\frac{3}{4}}}$ .

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Step 6.1

Multiply $y - x$ by $\frac{1}{x^{\frac{3}{4}}}$ .

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

Step 6.2

Multiply $y - x$ by $\frac{1}{x^{\frac{3}{4}}}$ .

$\frac{y - x}{x^{\frac{3}{4}}} d x + 4 x d y = 0$

Step 6.3

Multiply $4 x$ by $\frac{1}{x^{\frac{3}{4}}}$ .

$\frac{y - x}{x^{\frac{3}{4}}} d x + 4 x \frac{1}{x^{\frac{3}{4}}} d y = 0$

Step 6.4

Multiply $4 x \frac{1}{x^{\frac{3}{4}}}$ .

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Step 6.4.1

Combine $4$ and $\frac{1}{x^{\frac{3}{4}}}$ .

$\frac{y - x}{x^{\frac{3}{4}}} d x + x \frac{4}{x^{\frac{3}{4}}} d y = 0$

Step 6.4.2

Combine $x$ and $\frac{4}{x^{\frac{3}{4}}}$ .

$\frac{y - x}{x^{\frac{3}{4}}} d x + \frac{x \cdot 4}{x^{\frac{3}{4}}} d y = 0$

Step 6.5

Move $x^{\frac{3}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{y - x}{x^{\frac{3}{4}}} d x + x \cdot 4 x^{- \frac{3}{4}} d y = 0$

Step 6.6

Multiply $x$ by $x^{- \frac{3}{4}}$ by adding the exponents.

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Step 6.6.1

Move $x^{- \frac{3}{4}}$ .

$\frac{y - x}{x^{\frac{3}{4}}} d x + x^{- \frac{3}{4}} x \cdot 4 d y = 0$

Step 6.6.2

Multiply $x^{- \frac{3}{4}}$ by $x$ .

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Step 6.6.2.1

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

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

Step 6.6.2.2

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

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

Step 6.6.3

Write $1$ as a fraction with a common denominator.

$\frac{y - x}{x^{\frac{3}{4}}} d x + x^{- \frac{3}{4} + \frac{4}{4}} \cdot 4 d y = 0$

Step 6.6.4

Combine the numerators over the common denominator.

$\frac{y - x}{x^{\frac{3}{4}}} d x + x^{\frac{- 3 + 4}{4}} \cdot 4 d y = 0$

Step 6.6.5

Add $- 3$ and $4$ .

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

Step 6.7

Move $4$ to the left of $x^{\frac{1}{4}}$ .

$\frac{y - x}{x^{\frac{3}{4}}} d x + 4 x^{\frac{1}{4}} d y = 0$

Step 7

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

$f (x, y) = \int 4 x^{\frac{1}{4}} d y$

Step 8

Integrate $N (x, y) = 4 x^{\frac{1}{4}}$ to find $f (x, y)$ .

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Step 8.1

Apply the constant rule.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [4 x^{\frac{1}{4}} y + g (x)] = \frac{y - x}{x^{\frac{3}{4}}}$

Step 11.2

By the Sum Rule, the derivative of $4 x^{\frac{1}{4}} y + g (x)$ with respect to $x$ is $\frac{d}{d x} [4 x^{\frac{1}{4}} y] + \frac{d}{d x} [g (x)]$ .

$\frac{d}{d x} [4 x^{\frac{1}{4}} y] + \frac{d}{d x} [g (x)] = \frac{y - x}{x^{\frac{3}{4}}}$

Step 11.3

Evaluate $\frac{d}{d x} [4 x^{\frac{1}{4}} y]$ .

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Step 11.3.1

Since $4 y$ is constant with respect to $x$ , the derivative of $4 x^{\frac{1}{4}} y$ with respect to $x$ is $4 y \frac{d}{d x} [x^{\frac{1}{4}}]$ .

$4 y \frac{d}{d x} [x^{\frac{1}{4}}] + \frac{d}{d x} [g (x)] = \frac{y - x}{x^{\frac{3}{4}}}$

Step 11.3.2

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

$4 y (\frac{1}{4} x^{\frac{1}{4} - 1}) + \frac{d}{d x} [g (x)] = \frac{y - x}{x^{\frac{3}{4}}}$

Step 11.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$4 y (\frac{1}{4} x^{\frac{1}{4} - 1 \cdot \frac{4}{4}}) + \frac{d}{d x} [g (x)] = \frac{y - x}{x^{\frac{3}{4}}}$

Step 11.3.4

Combine $- 1$ and $\frac{4}{4}$ .

$4 y (\frac{1}{4} x^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}}) + \frac{d}{d x} [g (x)] = \frac{y - x}{x^{\frac{3}{4}}}$

Step 11.3.5

Combine the numerators over the common denominator.

$4 y (\frac{1}{4} x^{\frac{1 - 1 \cdot 4}{4}}) + \frac{d}{d x} [g (x)] = \frac{y - x}{x^{\frac{3}{4}}}$

Step 11.3.6

Simplify the numerator.

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Step 11.3.6.1

Multiply $- 1$ by $4$ .

$4 y (\frac{1}{4} x^{\frac{1 - 4}{4}}) + \frac{d}{d x} [g (x)] = \frac{y - x}{x^{\frac{3}{4}}}$

Step 11.3.6.2

Subtract $4$ from $1$ .

$4 y (\frac{1}{4} x^{\frac{- 3}{4}}) + \frac{d}{d x} [g (x)] = \frac{y - x}{x^{\frac{3}{4}}}$

Step 11.3.7

Move the negative in front of the fraction.

$4 y (\frac{1}{4} x^{- \frac{3}{4}}) + \frac{d}{d x} [g (x)] = \frac{y - x}{x^{\frac{3}{4}}}$

Step 11.3.8

Combine $\frac{1}{4}$ and $x^{- \frac{3}{4}}$ .

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

Step 11.3.9

Combine $4$ and $\frac{x^{- \frac{3}{4}}}{4}$ .

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

Step 11.3.10

Combine $y$ and $\frac{4 x^{- \frac{3}{4}}}{4}$ .

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

Step 11.3.11

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

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

Step 11.3.12

Move $x^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11.3.13

Cancel the common factor.

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

Step 11.3.14

Rewrite the expression.

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

Simplify $\frac{d g}{d x} + \frac{y}{x^{\frac{3}{4}}} - \frac{y - x}{x^{\frac{3}{4}}}$ .

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Step 12.1.1.1

Combine the numerators over the common denominator.

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

Step 12.1.1.2

Simplify each term.

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Step 12.1.1.2.1

Apply the distributive property.

$\frac{d g}{d x} + \frac{y - y - - x}{x^{\frac{3}{4}}} = 0$

Step 12.1.1.2.2

Multiply $- - x$ .

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Step 12.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{d g}{d x} + \frac{y - y + 1 x}{x^{\frac{3}{4}}} = 0$

Step 12.1.1.2.2.2

Multiply $x$ by $1$ .

$\frac{d g}{d x} + \frac{y - y + x}{x^{\frac{3}{4}}} = 0$

Step 12.1.1.3

Simplify by adding terms.

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Step 12.1.1.3.1

Subtract $y$ from $y$ .

$\frac{d g}{d x} + \frac{0 + x}{x^{\frac{3}{4}}} = 0$

Step 12.1.1.3.2

Add $0$ and $x$ .

$\frac{d g}{d x} + \frac{x}{x^{\frac{3}{4}}} = 0$

Step 12.1.1.4

Simplify each term.

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Step 12.1.1.4.1

Move $x^{\frac{3}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{d g}{d x} + x \cdot x^{- \frac{3}{4}} = 0$

Step 12.1.1.4.2

Multiply $x$ by $x^{- \frac{3}{4}}$ by adding the exponents.

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Step 12.1.1.4.2.1

Multiply $x$ by $x^{- \frac{3}{4}}$ .

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Step 12.1.1.4.2.1.1

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

$\frac{d g}{d x} + x^{1} x^{- \frac{3}{4}} = 0$

Step 12.1.1.4.2.1.2

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

$\frac{d g}{d x} + x^{1 - \frac{3}{4}} = 0$

Step 12.1.1.4.2.2

Write $1$ as a fraction with a common denominator.

$\frac{d g}{d x} + x^{\frac{4}{4} - \frac{3}{4}} = 0$

Step 12.1.1.4.2.3

Combine the numerators over the common denominator.

$\frac{d g}{d x} + x^{\frac{4 - 3}{4}} = 0$

Step 12.1.1.4.2.4

Subtract $3$ from $4$ .

$\frac{d g}{d x} + x^{\frac{1}{4}} = 0$

Step 12.1.2

Find a common factor $x^{\frac{1}{4}}$ that is present in each term.

${({(\frac{d g}{d x})}^{\frac{1}{4}})}^{4} + x^{\frac{1}{4}}$

Step 12.1.3

Substitute $u$ for $x^{\frac{1}{4}}$ .

${({(\frac{d g}{d x})}^{\frac{1}{4}})}^{4} + u = 0$

Step 12.1.4

Solve for $u$ .

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Step 12.1.4.1

Simplify each term.

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Step 12.1.4.1.1

Multiply the exponents in ${({(\frac{d g}{d x})}^{\frac{1}{4}})}^{4}$ .

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Step 12.1.4.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(\frac{d g}{d x})}^{\frac{1}{4} \cdot 4} + u = 0$

Step 12.1.4.1.1.2

Cancel the common factor of $4$ .

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Step 12.1.4.1.1.2.1

Cancel the common factor.

${(\frac{d g}{d x})}^{\frac{1}{4} \cdot 4} + u = 0$

Step 12.1.4.1.1.2.2

Rewrite the expression.

${(\frac{d g}{d x})}^{1} + u = 0$

Step 12.1.4.1.2

Simplify.

$\frac{d g}{d x} + u = 0$

Step 12.1.4.2

Subtract $\frac{d g}{d x}$ from both sides of the equation.

$u = - \frac{d g}{d x}$

Step 12.1.5

Substitute $\frac{d g}{d x}$ for $u$ .

$x^{\frac{1}{4}} = - \frac{d g}{d x}$

Step 12.1.6

Subtract $x^{\frac{1}{4}}$ from both sides of the equation.

$\frac{d g}{d x} = - x^{\frac{1}{4}}$

Step 13

Find the antiderivative of $- x^{\frac{1}{4}}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = - x^{\frac{1}{4}}$ .

$\int \frac{d g}{d x} d x = \int - x^{\frac{1}{4}} d x$

Step 13.2

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

$g (x) = \int - x^{\frac{1}{4}} d x$

Step 13.3

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

$g (x) = - \int x^{\frac{1}{4}} d x$

Step 13.4

By the Power Rule, the integral of $x^{\frac{1}{4}}$ with respect to $x$ is $\frac{4}{5} x^{\frac{5}{4}}$ .

$g (x) = - (\frac{4}{5} x^{\frac{5}{4}} + C)$

Step 13.5

Rewrite $- (\frac{4}{5} x^{\frac{5}{4}} + C)$ as $- \frac{4}{5} x^{\frac{5}{4}} + C$ .

$g (x) = - \frac{4}{5} x^{\frac{5}{4}} + C$

Step 14

Substitute for $g (x)$ in $f (x, y) = 4 x^{\frac{1}{4}} y + g (x)$ .

$f (x, y) = 4 x^{\frac{1}{4}} y - \frac{4}{5} x^{\frac{5}{4}} + C$

Step 15

Simplify $f (x, y) = 4 x^{\frac{1}{4}} y - \frac{4}{5} x^{\frac{5}{4}} + C$ .

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Step 15.1

Simplify each term.

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Step 15.1.1

Combine $x^{\frac{5}{4}}$ and $\frac{4}{5}$ .

$f (x, y) = 4 x^{\frac{1}{4}} y - \frac{x^{\frac{5}{4}} \cdot 4}{5} + C$

Step 15.1.2

Move $4$ to the left of $x^{\frac{5}{4}}$ .

$f (x, y) = 4 x^{\frac{1}{4}} y - \frac{4 x^{\frac{5}{4}}}{5} + C$

Step 15.2

Reorder factors in $f (x, y) = 4 x^{\frac{1}{4}} y - \frac{4 x^{\frac{5}{4}}}{5} + C$ .

$f (x, y) = 4 y x^{\frac{1}{4}} - \frac{4 x^{\frac{5}{4}}}{5} + C$