Calculus Examples

$(5 x + 3 e^{y}) d x + 2 x e^{y} d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 5 x + 3 e^{y}$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [5 x + 3 e^{y}]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $5 x + 3 e^{y}$ with respect to $y$ is $\frac{d}{d y} [5 x] + \frac{d}{d y} [3 e^{y}]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [5 x] + \frac{d}{d y} [3 e^{y}]$

Step 1.2.2

Since $5 x$ is constant with respect to $y$ , the derivative of $5 x$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [3 e^{y}]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 0 + 3 \frac{d}{d y} [e^{y}]$

Step 1.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d y} [a^{y}]$ is $a^{y} \ln (a)$ where $a$ = $e$ .

$\frac{\partial M}{\partial y} = 0 + 3 e^{y}$

Step 1.4

Add $0$ and $3 e^{y}$ .

$\frac{\partial M}{\partial y} = 3 e^{y}$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $1$ .

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

Step 3

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

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Step 3.1

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

$3 e^{y} = 2 e^{y}$

Step 3.2

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

$3 e^{y} = 2 e^{y}$ 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 $3 e^{y}$ for $\frac{\partial M}{\partial y}$ .

$\frac{3 e^{y} - \frac{\partial N}{\partial x}}{N}$

Step 4.2

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

$\frac{3 e^{y} - (2 e^{y})}{N}$

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $e^{y}$ out of $3 e^{y} - 1 \cdot 2 e^{y}$ .

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Step 4.3.2.1.1

Factor $e^{y}$ out of $3 e^{y}$ .

$\frac{e^{y} \cdot 3 - 1 \cdot 2 e^{y}}{2 x e^{y}}$

Step 4.3.2.1.2

Factor $e^{y}$ out of $- 1 \cdot 2 e^{y}$ .

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

Step 4.3.2.1.3

Factor $e^{y}$ out of $e^{y} \cdot 3 + e^{y} (- 1 \cdot 2)$ .

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

Step 4.3.2.2

Multiply $- 1$ by $2$ .

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

Step 4.3.2.3

Subtract $2$ from $3$ .

$\frac{e^{y} \cdot 1}{2 x e^{y}}$

Step 4.3.3

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

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

Step 4.3.4

Cancel the common factor of $e^{y}$ .

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Step 4.3.4.1

Cancel the common factor.

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

Step 4.3.4.2

Rewrite the expression.

$\frac{1}{2 x}$

Step 4.4

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

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

Step 5

Evaluate the integral $e^{\int \frac{1}{2 x} d x}$ .

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Step 5.1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 5.2

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

$μ (x, y) = e^{\frac{1}{2} (\ln (| x |) + C)}$

Step 5.3

Simplify.

$μ (x, y) = e^{\frac{1}{2} \ln (| x |)} + C$

Step 5.4

Simplify each term.

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Step 5.4.1

Simplify $\frac{1}{2} \ln (| x |)$ by moving $\frac{1}{2}$ inside the logarithm.

$μ (x, y) = e^{\ln ({| x |}^{\frac{1}{2}})} + C$

Step 5.4.2

Exponentiation and log are inverse functions.

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

Step 6

Multiply both sides of $(5 x + 3 e^{y}) d x + 2 x e^{y} d y = 0$ by the integration factor $x^{\frac{1}{2}}$ .

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Step 6.1

Multiply $5 x + 3 e^{y}$ by $x^{\frac{1}{2}}$ .

$(5 x + 3 e^{y}) x^{\frac{1}{2}} d x + 2 x e^{y} d y = 0$

Step 6.2

Apply the distributive property.

$(5 x \cdot x^{\frac{1}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x e^{y} d y = 0$

Step 6.3

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 6.3.1

Move $x^{\frac{1}{2}}$ .

$(5 (x^{\frac{1}{2}} x) + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x e^{y} d y = 0$

Step 6.3.2

Multiply $x^{\frac{1}{2}}$ by $x$ .

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Step 6.3.2.1

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

$(5 (x^{\frac{1}{2}} x^{1}) + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x e^{y} d y = 0$

Step 6.3.2.2

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

$(5 x^{\frac{1}{2} + 1} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x e^{y} d y = 0$

Step 6.3.3

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

$(5 x^{\frac{1}{2} + \frac{2}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x e^{y} d y = 0$

Step 6.3.4

Combine the numerators over the common denominator.

$(5 x^{\frac{1 + 2}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x e^{y} d y = 0$

Step 6.3.5

Add $1$ and $2$ .

$(5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x e^{y} d y = 0$

Step 6.4

Multiply $2 x e^{y}$ by $x^{\frac{1}{2}}$ .

$(5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x e^{y} x^{\frac{1}{2}} d y = 0$

Step 6.5

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 6.5.1

Move $x^{\frac{1}{2}}$ .

$(5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 (x^{\frac{1}{2}} x) e^{y} d y = 0$

Step 6.5.2

Multiply $x^{\frac{1}{2}}$ by $x$ .

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Step 6.5.2.1

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

$(5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 (x^{\frac{1}{2}} x^{1}) e^{y} d y = 0$

Step 6.5.2.2

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

$(5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x^{\frac{1}{2} + 1} e^{y} d y = 0$

Step 6.5.3

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

$(5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x^{\frac{1}{2} + \frac{2}{2}} e^{y} d y = 0$

Step 6.5.4

Combine the numerators over the common denominator.

$(5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x^{\frac{1 + 2}{2}} e^{y} d y = 0$

Step 6.5.5

Add $1$ and $2$ .

$(5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}) d x + 2 x^{\frac{3}{2}} e^{y} d y = 0$

Step 7

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

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

Step 8

Integrate $N (x, y) = 2 x^{\frac{3}{2}} e^{y}$ to find $f (x, y)$ .

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Step 8.1

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

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

Step 8.2

The integral of $e^{y}$ with respect to $y$ is $e^{y}$ .

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

Step 8.3

Simplify.

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

Step 9

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

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

Step 10

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

$\frac{\partial f}{\partial x} = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [2 x^{\frac{3}{2}} e^{y} + g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.2

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

$\frac{d}{d x} [2 x^{\frac{3}{2}} e^{y}] + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3

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

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Step 11.3.1

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

$2 e^{y} \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$2 e^{y} (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 e^{y} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.4

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

$2 e^{y} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.5

Combine the numerators over the common denominator.

$2 e^{y} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.6

Simplify the numerator.

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Step 11.3.6.1

Multiply $- 1$ by $2$ .

$2 e^{y} (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.6.2

Subtract $2$ from $3$ .

$2 e^{y} (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.7

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$2 e^{y} \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.8

Combine $\frac{3 x^{\frac{1}{2}}}{2}$ and $2$ .

$\frac{3 x^{\frac{1}{2}} \cdot 2}{2} e^{y} + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.9

Multiply $2$ by $3$ .

$\frac{6 x^{\frac{1}{2}}}{2} e^{y} + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.10

Combine $\frac{6 x^{\frac{1}{2}}}{2}$ and $e^{y}$ .

$\frac{6 x^{\frac{1}{2}} e^{y}}{2} + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.11

Factor $2$ out of $6 x^{\frac{1}{2}} e^{y}$ .

$\frac{2 (3 x^{\frac{1}{2}} e^{y})}{2} + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.12

Cancel the common factors.

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Step 11.3.12.1

Factor $2$ out of $2$ .

$\frac{2 (3 x^{\frac{1}{2}} e^{y})}{2 (1)} + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.12.2

Cancel the common factor.

$\frac{2 (3 x^{\frac{1}{2}} e^{y})}{2 \cdot 1} + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.12.3

Rewrite the expression.

$\frac{3 x^{\frac{1}{2}} e^{y}}{1} + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.3.12.4

Divide $3 x^{\frac{1}{2}} e^{y}$ by $1$ .

$3 x^{\frac{1}{2}} e^{y} + \frac{d}{d x} [g (x)] = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.4

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

$3 x^{\frac{1}{2}} e^{y} + \frac{d g}{d x} = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

Step 11.5

Reorder terms.

$\frac{d g}{d x} + 3 e^{y} x^{\frac{1}{2}} = 5 x^{\frac{3}{2}} + 3 e^{y} x^{\frac{1}{2}}$

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

Find a common factor $x^{\frac{3}{2}}$ that is present in each term.

${({(\frac{d g}{d x})}^{\frac{3}{2}})}^{\frac{2}{3}} + 3 {(e x^{\frac{3}{2}})}^{\frac{2 y}{3} + \frac{1}{3}} - 5 x^{\frac{3}{2}} - 3 {(e x^{\frac{3}{2}})}^{\frac{2 y}{3} + \frac{1}{3}}$

Step 12.1.2

Substitute $u$ for $x^{\frac{3}{2}}$ .

${({(\frac{d g}{d x})}^{\frac{3}{2}})}^{\frac{2}{3}} + 3 {(e (u))}^{\frac{2 y}{3} + \frac{1}{3}} - 5 u - 3 {(e (u))}^{\frac{2 y}{3} + \frac{1}{3}} = 0$

Step 12.1.3

Solve for $u$ .

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Step 12.1.3.1

Simplify ${({(\frac{d g}{d x})}^{\frac{3}{2}})}^{\frac{2}{3}} + 3 {(e u)}^{\frac{2 y}{3} + \frac{1}{3}} - 5 u - 3 {(e u)}^{\frac{2 y}{3} + \frac{1}{3}}$ .

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Step 12.1.3.1.1

Combine the opposite terms in ${({(\frac{d g}{d x})}^{\frac{3}{2}})}^{\frac{2}{3}} + 3 {(e u)}^{\frac{2 y}{3} + \frac{1}{3}} - 5 u - 3 {(e u)}^{\frac{2 y}{3} + \frac{1}{3}}$ .

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Step 12.1.3.1.1.1

Subtract $3 {(e u)}^{\frac{2 y}{3} + \frac{1}{3}}$ from $3 {(e u)}^{\frac{2 y}{3} + \frac{1}{3}}$ .

${({(\frac{d g}{d x})}^{\frac{3}{2}})}^{\frac{2}{3}} - 5 u + 0 = 0$

Step 12.1.3.1.1.2

Add ${({(\frac{d g}{d x})}^{\frac{3}{2}})}^{\frac{2}{3}} - 5 u$ and $0$ .

${({(\frac{d g}{d x})}^{\frac{3}{2}})}^{\frac{2}{3}} - 5 u = 0$

Step 12.1.3.1.2

Simplify each term.

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Step 12.1.3.1.2.1

Multiply the exponents in ${({(\frac{d g}{d x})}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 12.1.3.1.2.1.1

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

${(\frac{d g}{d x})}^{\frac{3}{2} \cdot \frac{2}{3}} - 5 u = 0$

Step 12.1.3.1.2.1.2

Cancel the common factor of $3$ .

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Step 12.1.3.1.2.1.2.1

Cancel the common factor.

${(\frac{d g}{d x})}^{\frac{3}{2} \cdot \frac{2}{3}} - 5 u = 0$

Step 12.1.3.1.2.1.2.2

Rewrite the expression.

${(\frac{d g}{d x})}^{\frac{1}{2} \cdot 2} - 5 u = 0$

Step 12.1.3.1.2.1.3

Cancel the common factor of $2$ .

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Step 12.1.3.1.2.1.3.1

Cancel the common factor.

${(\frac{d g}{d x})}^{\frac{1}{2} \cdot 2} - 5 u = 0$

Step 12.1.3.1.2.1.3.2

Rewrite the expression.

${(\frac{d g}{d x})}^{1} - 5 u = 0$

Step 12.1.3.1.2.2

Simplify.

$\frac{d g}{d x} - 5 u = 0$

Step 12.1.3.2

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

$- 5 u = - \frac{d g}{d x}$

Step 12.1.3.3

Divide each term in $- 5 u = - \frac{d g}{d x}$ by $- 5$ and simplify.

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Step 12.1.3.3.1

Divide each term in $- 5 u = - \frac{d g}{d x}$ by $- 5$ .

$\frac{- 5 u}{- 5} = \frac{- \frac{d g}{d x}}{- 5}$

Step 12.1.3.3.2

Simplify the left side.

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Step 12.1.3.3.2.1

Cancel the common factor of $- 5$ .

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Step 12.1.3.3.2.1.1

Cancel the common factor.

$\frac{- 5 u}{- 5} = \frac{- \frac{d g}{d x}}{- 5}$

Step 12.1.3.3.2.1.2

Divide $u$ by $1$ .

$u = \frac{- \frac{d g}{d x}}{- 5}$

Step 12.1.3.3.3

Simplify the right side.

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Step 12.1.3.3.3.1

Dividing two negative values results in a positive value.

$u = \frac{\frac{d g}{d x}}{5}$

Step 12.1.4

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

$x^{\frac{3}{2}} = \frac{\frac{d g}{d x}}{5}$

Step 13

Find the antiderivative of $\frac{\frac{d g}{d x}}{5}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $x^{\frac{3}{2}} = \frac{\frac{d g}{d x}}{5}$ .

$\int x^{\frac{3}{2}} d x = \int \frac{\frac{d g}{d x}}{5} d x$

Step 13.2

Evaluate $\int x^{\frac{3}{2}} d x$ .

$g (x) = \int \frac{\frac{d g}{d x}}{5} d x$

Step 13.3

Apply the constant rule.

$g (x) = \frac{\frac{d g}{d x}}{5} x + C$

Step 14

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

$f (x, y) = 2 x^{\frac{3}{2}} e^{y} + \frac{\frac{d g}{d x}}{5} x + C$

Step 15

Simplify $f (x, y) = 2 x^{\frac{3}{2}} e^{y} + \frac{\frac{d g}{d x}}{5} x + C$ .

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Step 15.1

Combine $\frac{\frac{d g}{d x}}{5}$ and $x$ .

$f (x, y) = 2 x^{\frac{3}{2}} e^{y} + \frac{\frac{d g}{d x} x}{5} + C$

Step 15.2

Reorder factors in $f (x, y) = 2 x^{\frac{3}{2}} e^{y} + \frac{\frac{d g}{d x} x}{5} + C$ .

$f (x, y) = 2 x^{\frac{3}{2}} e^{y} + \frac{x \frac{d g}{d x}}{5} + C$