Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

Since $e^{x}$ is constant with respect to $y$ , the derivative of $e^{x}$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [y^{2}]$

Step 1.4

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

$\frac{\partial M}{\partial y} = 0 + 2 y$

Step 1.5

Add $0$ and $2 y$ .

$\frac{\partial M}{\partial y} = 2 y$

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $y$ by $1$ .

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

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.5.1

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

$\frac{\partial N}{\partial x} = y - \frac{e^{x}}{y} + 0$

Step 2.5.2

Add $y - \frac{e^{x}}{y}$ and $0$ .

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

$2 y = y - \frac{e^{x}}{y}$ is not an identity.

Step 4

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

Tap for more steps...

Step 4.1

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

$\frac{y - \frac{e^{x}}{y} - \frac{\partial M}{\partial y}}{M}$

Step 4.2

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

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

Step 4.3

Substitute $e^{x} + y^{2}$ for $M$ .

Tap for more steps...

Step 4.3.1

Substitute $e^{x} + y^{2}$ for $M$ .

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

Step 4.3.2

Multiply the numerator and denominator of the fraction by $y$ .

Tap for more steps...

Step 4.3.2.1

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

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

Step 4.3.2.2

Combine.

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

Step 4.3.3

Apply the distributive property.

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

Step 4.3.4

Cancel the common factor of $y$ .

Tap for more steps...

Step 4.3.4.1

Move the leading negative in $- \frac{e^{x}}{y}$ into the numerator.

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

Step 4.3.4.2

Cancel the common factor.

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

Step 4.3.4.3

Rewrite the expression.

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

Step 4.3.5

Simplify the numerator.

Tap for more steps...

Step 4.3.5.1

Multiply $y$ by $y$ .

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

Step 4.3.5.2

Rewrite using the commutative property of multiplication.

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

Step 4.3.5.3

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

Tap for more steps...

Step 4.3.5.3.1

Move $y$ .

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

Step 4.3.5.3.2

Multiply $y$ by $y$ .

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

Step 4.3.5.4

Multiply $- 1$ by $2$ .

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

Step 4.3.5.5

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

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

Step 4.3.6

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

Tap for more steps...

Step 4.3.6.1

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

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

Step 4.3.6.2

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

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

Step 4.3.6.3

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

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

Step 4.3.7

Cancel the common factor of $- y^{2} - e^{x}$ and $e^{x} + y^{2}$ .

Tap for more steps...

Step 4.3.7.1

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

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

Step 4.3.7.2

Factor $- 1$ out of $- e^{x}$ .

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

Step 4.3.7.3

Factor $- 1$ out of $- (y^{2}) - (e^{x})$ .

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

Step 4.3.7.4

Rewrite $- (y^{2} + e^{x})$ as $- 1 (y^{2} + e^{x})$ .

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

Step 4.3.7.5

Reorder terms.

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

Step 4.3.7.6

Cancel the common factor.

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

Step 4.3.7.7

Rewrite the expression.

$\frac{- 1}{y}$

Step 4.3.8

Substitute $e^{x} + y^{2}$ for $M$ .

$- \frac{1}{y}$

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 $e^{\int - \frac{1}{y} d y}$ .

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

$μ (x, y) = e^{- (\ln (| y |) + C)}$

Step 5.3

Simplify.

$μ (x, y) = e^{- \ln (| y |)} + C$

Step 5.4

Simplify each term.

Tap for more steps...

Step 5.4.1

Simplify $- \ln (| y |)$ by moving $- 1$ inside the logarithm.

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

Step 5.4.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| y |}^{- 1} + C$

Step 5.4.3

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

$μ (x, y) = \frac{1}{| y |} + C$

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Simplify the numerator.

Tap for more steps...

Step 6.5.1

To write $x y$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 6.5.2

Combine the numerators over the common denominator.

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

Step 6.5.3

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

Tap for more steps...

Step 6.5.3.1

Move $y$ .

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

Step 6.5.3.2

Multiply $y$ by $y$ .

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

Step 6.5.4

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

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

Step 6.5.5

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

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

Step 6.5.6

Combine the numerators over the common denominator.

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

Step 6.5.7

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

Tap for more steps...

Step 6.5.7.1

Move $y$ .

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

Step 6.5.7.2

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

Tap for more steps...

Step 6.5.7.2.1

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

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

Step 6.5.7.2.2

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

$\frac{e^{x} + y^{2}}{y} d x + \frac{\frac{x y^{2} - e^{x} - 2 y^{1 + 2}}{y}}{y} d y = 0$

Step 6.5.7.3

Add $1$ and $2$ .

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

Step 6.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.7

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

Tap for more steps...

Step 6.7.1

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

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

Step 6.7.2

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

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

Step 6.7.3

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

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

Step 6.7.4

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

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

Step 6.7.5

Add $1$ and $1$ .

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

Step 7

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

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

Step 8

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

Tap for more steps...

Step 8.1

Split the fraction into multiple fractions.

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

Step 8.2

Split the single integral into multiple integrals.

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

Step 8.3

Cancel the common factor of $y^{2}$ and $y$ .

Tap for more steps...

Step 8.3.1

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

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

Step 8.3.2

Cancel the common factors.

Tap for more steps...

Step 8.3.2.1

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

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

Step 8.3.2.2

Factor $y$ out of $y^{1}$ .

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

Step 8.3.2.3

Cancel the common factor.

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

Step 8.3.2.4

Rewrite the expression.

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

Step 8.3.2.5

Divide $y$ by $1$ .

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

Apply the constant rule.

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

Step 8.7

Simplify.

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

Step 9

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

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

Step 10

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

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

Step 11

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

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

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

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

Step 11.3

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

Tap for more steps...

Step 11.3.1

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

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

Step 11.3.2

Rewrite $\frac{1}{y}$ as $y^{- 1}$ .

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

Step 11.3.3

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

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

Step 11.4

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

Tap for more steps...

Step 11.4.1

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

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

Step 11.4.2

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

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

Step 11.4.3

Multiply $x$ by $1$ .

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

Step 11.5

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

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

Step 11.6

Simplify.

Tap for more steps...

Step 11.6.1

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

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

Step 11.6.2

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

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

Step 11.6.3

Reorder terms.

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

Step 12

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

Tap for more steps...

Step 12.1

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

Tap for more steps...

Step 12.1.1

Move all terms containing variables to the left side of the equation.

Tap for more steps...

Step 12.1.1.1

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

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

Step 12.1.1.2

Combine the numerators over the common denominator.

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

Step 12.1.1.3

Simplify each term.

Tap for more steps...

Step 12.1.1.3.1

Apply the distributive property.

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

Step 12.1.1.3.2

Simplify.

Tap for more steps...

Step 12.1.1.3.2.1

Multiply $- - e^{x}$ .

Tap for more steps...

Step 12.1.1.3.2.1.1

Multiply $- 1$ by $- 1$ .

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

Step 12.1.1.3.2.1.2

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

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

Step 12.1.1.3.2.2

Multiply $- 2$ by $- 1$ .

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

Step 12.1.1.4

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

Tap for more steps...

Step 12.1.1.4.1

Add $- e^{x}$ and $e^{x}$ .

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

Step 12.1.1.4.2

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

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

Step 12.1.1.5

Simplify each term.

Tap for more steps...

Step 12.1.1.5.1

Factor $y^{2}$ out of $- x y^{2} + 2 y^{3}$ .

Tap for more steps...

Step 12.1.1.5.1.1

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

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

Step 12.1.1.5.1.2

Factor $y^{2}$ out of $2 y^{3}$ .

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

Step 12.1.1.5.1.3

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

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

Step 12.1.1.5.2

Cancel the common factor of $y^{2}$ .

Tap for more steps...

Step 12.1.1.5.2.1

Cancel the common factor.

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

Step 12.1.1.5.2.2

Divide $- x + 2 y$ by $1$ .

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

Step 12.1.1.6

Combine the opposite terms in $\frac{d g}{d y} + x - x + 2 y$ .

Tap for more steps...

Step 12.1.1.6.1

Subtract $x$ from $x$ .

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

Step 12.1.1.6.2

Add $\frac{d g}{d y}$ and $0$ .

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

Step 12.1.2

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

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

Step 13

Find the antiderivative of $- 2 y$ to find $g (y)$ .

Tap for more steps...

Step 13.1

Integrate both sides of $\frac{d g}{d y} = - 2 y$ .

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

Step 13.2

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

$g (y) = \int - 2 y d y$

Step 13.3

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

$g (y) = - 2 \int y d y$

Step 13.4

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

$g (y) = - 2 (\frac{1}{2} y^{2} + C)$

Step 13.5

Simplify the answer.

Tap for more steps...

Step 13.5.1

Rewrite $- 2 (\frac{1}{2} y^{2} + C)$ as $- 2 (\frac{1}{2}) y^{2} + C$ .

$g (y) = - 2 (\frac{1}{2}) y^{2} + C$

Step 13.5.2

Simplify.

Tap for more steps...

Step 13.5.2.1

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

$g (y) = \frac{- 2}{2} y^{2} + C$

Step 13.5.2.2

Cancel the common factor of $- 2$ and $2$ .

Tap for more steps...

Step 13.5.2.2.1

Factor $2$ out of $- 2$ .

$g (y) = \frac{2 \cdot - 1}{2} y^{2} + C$

Step 13.5.2.2.2

Cancel the common factors.

Tap for more steps...

Step 13.5.2.2.2.1

Factor $2$ out of $2$ .

$g (y) = \frac{2 \cdot - 1}{2 (1)} y^{2} + C$

Step 13.5.2.2.2.2

Cancel the common factor.

$g (y) = \frac{2 \cdot - 1}{2 \cdot 1} y^{2} + C$

Step 13.5.2.2.2.3

Rewrite the expression.

$g (y) = \frac{- 1}{1} y^{2} + C$

Step 13.5.2.2.2.4

Divide $- 1$ by $1$ .

$g (y) = - y^{2} + C$

Step 14

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

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