Calculus Examples

$(y^{2} + x y^{3}) d x + (5 y^{2} - x y + y^{3} \sin (y)) d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 2 y + x (3 y^{2})$

Step 1.3.3

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

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

Step 1.4

Reorder terms.

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

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = 5 y^{2} - x y + y^{3} \sin (y)$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [5 y^{2} - x y + y^{3} \sin (y)]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $5 y^{2} - x y + y^{3} \sin (y)$ with respect to $x$ is $\frac{d}{d x} [5 y^{2}] + \frac{d}{d x} [- x y] + \frac{d}{d x} [y^{3} \sin (y)]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [5 y^{2}] + \frac{d}{d x} [- x y] + \frac{d}{d x} [y^{3} \sin (y)]$

Step 2.2.2

Since $5 y^{2}$ is constant with respect to $x$ , the derivative of $5 y^{2}$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = 0 + \frac{d}{d x} [- x y] + \frac{d}{d x} [y^{3} \sin (y)]$

Step 2.3

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

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Step 2.3.1

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

$\frac{\partial N}{\partial x} = 0 - y \frac{d}{d x} [x] + \frac{d}{d x} [y^{3} \sin (y)]$

Step 2.3.2

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

$\frac{\partial N}{\partial x} = 0 - y \cdot 1 + \frac{d}{d x} [y^{3} \sin (y)]$

Step 2.3.3

Multiply $- 1$ by $1$ .

$\frac{\partial N}{\partial x} = 0 - y + \frac{d}{d x} [y^{3} \sin (y)]$

Step 2.4

Since $y^{3} \sin (y)$ is constant with respect to $x$ , the derivative of $y^{3} \sin (y)$ with respect to $x$ is $0$ .

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

Step 2.5

Combine terms.

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Step 2.5.1

Subtract $y$ from $0$ .

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

Step 2.5.2

Add $- y$ and $0$ .

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

Step 3

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

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Step 3.1

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

$3 y^{2} x + 2 y = - y$

Step 3.2

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

$3 y^{2} x + 2 y = - y$ is not an identity.

Step 4

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

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Step 4.1

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

$\frac{- y - \frac{\partial M}{\partial y}}{M}$

Step 4.2

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

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

Step 4.3

Substitute $y^{2} + x y^{3}$ for $M$ .

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Step 4.3.1

Substitute $y^{2} + x y^{3}$ for $M$ .

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

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

Step 4.3.2.2

Multiply $3$ by $- 1$ .

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

Step 4.3.2.3

Multiply $2$ by $- 1$ .

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

Step 4.3.2.4

Subtract $2 y$ from $- y$ .

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

Step 4.3.2.5

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

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Step 4.3.2.5.1

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

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

Step 4.3.2.5.2

Factor $3 y$ out of $- 3 y$ .

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

Step 4.3.2.5.3

Factor $3 y$ out of $3 y (- y x) + 3 y (- 1)$ .

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

Step 4.3.3

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

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Step 4.3.3.1

Multiply by $1$ .

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

Step 4.3.3.2

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

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

Step 4.3.3.3

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

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

Step 4.3.4

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

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Step 4.3.4.1

Factor $y$ out of $3 y (- y x - 1)$ .

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

Step 4.3.4.2

Cancel the common factors.

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Step 4.3.4.2.1

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

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

Step 4.3.4.2.2

Cancel the common factor.

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

Step 4.3.4.2.3

Rewrite the expression.

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

Step 4.3.5

Reorder the terms.

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

Step 4.3.6

Cancel the common factor of $- x y - 1$ and $x y + 1$ .

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Step 4.3.6.1

Factor $- 1$ out of $- x y$ .

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

Step 4.3.6.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 4.3.6.3

Factor $- 1$ out of $- (x y) - 1 (1)$ .

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

Step 4.3.6.4

Rewrite $- (x y + 1)$ as $- 1 (x y + 1)$ .

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

Step 4.3.6.5

Cancel the common factor.

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

Step 4.3.6.6

Rewrite the expression.

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

Step 4.3.7

Multiply $3$ by $- 1$ .

$\frac{- 3}{y}$

Step 4.3.8

Substitute $y^{2} + x y^{3}$ for $M$ .

$- \frac{3}{y}$

Step 4.4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

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

Step 5.3

Multiply $3$ by $- 1$ .

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

Step 5.4

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

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

Step 5.5

Simplify.

$μ (x, y) = e^{- 3 \ln (| y |)} + C$

Step 5.6

Simplify each term.

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Step 5.6.1

Simplify $- 3 \ln (| y |)$ by moving $- 3$ inside the logarithm.

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

Step 5.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| y |}^{- 3} + C$

Step 5.6.3

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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Step 6.3.1

Multiply by $1$ .

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

Step 6.3.2

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

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

Step 6.3.3

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

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

Step 6.4

Cancel the common factors.

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Step 6.4.1

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

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

Step 6.4.2

Cancel the common factor.

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

Step 6.4.3

Rewrite the expression.

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

Step 6.5

Multiply $5 y^{2} - x y + y^{3} \sin (y)$ by $\frac{1}{y^{3}}$ .

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

Step 6.6

Apply the distributive property.

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

Step 6.7

Simplify.

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Step 6.7.1

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

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Step 6.7.1.1

Factor $y^{2}$ out of $5 y^{2}$ .

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

Step 6.7.1.2

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

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

Step 6.7.1.3

Cancel the common factor.

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

Step 6.7.1.4

Rewrite the expression.

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

Step 6.7.2

Combine $5$ and $\frac{1}{y}$ .

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

Step 6.7.3

Cancel the common factor of $y$ .

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Step 6.7.3.1

Factor $y$ out of $- x y$ .

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

Step 6.7.3.2

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

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

Step 6.7.3.3

Cancel the common factor.

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

Step 6.7.3.4

Rewrite the expression.

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

Step 6.7.4

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

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

Step 6.7.5

Cancel the common factor of $y^{3}$ .

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Step 6.7.5.1

Factor $y^{3}$ out of $y^{3} \sin (y)$ .

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

Step 6.7.5.2

Cancel the common factor.

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

Step 6.7.5.3

Rewrite the expression.

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

Step 7

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

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

Step 8

Integrate $M (x, y) = \frac{1 + x y}{y}$ to find $f (x, y)$ .

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Step 8.1

Split the fraction into multiple fractions.

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

Step 8.2

Split the single integral into multiple integrals.

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

Step 8.3

Cancel the common factor of $y$ .

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Step 8.3.1

Cancel the common factor.

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

Step 8.3.2

Divide $x$ by $1$ .

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

Step 8.4

Apply the constant rule.

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

Step 8.5

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

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

Step 8.6

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

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

Step 8.7

Simplify.

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

Step 9

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

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

Step 10

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

$\frac{\partial f}{\partial y} = \frac{5}{y} - \frac{x}{y^{2}} + \sin (y)$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.4

Since $\frac{1}{2} x^{2}$ is constant with respect to $y$ , the derivative of $\frac{1}{2} x^{2}$ with respect to $y$ is $0$ .

$x (- y^{- 2}) + 0 + \frac{d}{d y} [g (y)] = \frac{5}{y} - \frac{x}{y^{2}} + \sin (y)$

Step 11.5

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

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

Step 11.6

Simplify.

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Step 11.6.1

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

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

Step 11.6.2

Combine terms.

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Step 11.6.2.1

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

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

Step 11.6.2.2

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

$- \frac{x}{y^{2}} + \frac{d g}{d y} = \frac{5}{y} - \frac{x}{y^{2}} + \sin (y)$

Step 11.6.3

Reorder terms.

$\frac{d g}{d y} - \frac{x}{y^{2}} = \frac{5}{y} - \frac{x}{y^{2}} + \sin (y)$

Step 12

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

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Step 12.1

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

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Step 12.1.1

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

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Step 12.1.1.1

Subtract $\frac{5}{y}$ from both sides of the equation.

$\frac{d g}{d y} - \frac{x}{y^{2}} - \frac{5}{y} = - \frac{x}{y^{2}} + \sin (y)$

Step 12.1.1.2

Add $\frac{x}{y^{2}}$ to both sides of the equation.

$\frac{d g}{d y} - \frac{x}{y^{2}} - \frac{5}{y} + \frac{x}{y^{2}} = \sin (y)$

Step 12.1.1.3

Subtract $\sin (y)$ from both sides of the equation.

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

Step 12.1.1.4

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

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Step 12.1.1.4.1

Add $- \frac{x}{y^{2}}$ and $\frac{x}{y^{2}}$ .

$\frac{d g}{d y} - \frac{5}{y} + 0 - \sin (y) = 0$

Step 12.1.1.4.2

Add $\frac{d g}{d y} - \frac{5}{y}$ and $0$ .

$\frac{d g}{d y} - \frac{5}{y} - \sin (y) = 0$

Step 12.1.2

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

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Step 12.1.2.1

Add $\frac{5}{y}$ to both sides of the equation.

$\frac{d g}{d y} - \sin (y) = \frac{5}{y}$

Step 12.1.2.2

Add $\sin (y)$ to both sides of the equation.

$\frac{d g}{d y} = \frac{5}{y} + \sin (y)$

Step 13

Find the antiderivative of $\frac{5}{y} + \sin (y)$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = \frac{5}{y} + \sin (y)$ .

$\int \frac{d g}{d y} d y = \int \frac{5}{y} + \sin (y) d y$

Step 13.2

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

$g (y) = \int \frac{5}{y} + \sin (y) d y$

Step 13.3

Split the single integral into multiple integrals.

$g (y) = \int \frac{5}{y} d y + \int \sin (y) d y$

Step 13.4

Since $5$ is constant with respect to $y$ , move $5$ out of the integral.

$g (y) = 5 \int \frac{1}{y} d y + \int \sin (y) d y$

Step 13.5

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

$g (y) = 5 (\ln (| y |) + C) + \int \sin (y) d y$

Step 13.6

The integral of $\sin (y)$ with respect to $y$ is $- \cos (y)$ .

$g (y) = 5 (\ln (| y |) + C) - \cos (y) + C$

Step 13.7

Simplify.

$g (y) = 5 \ln (| y |) - \cos (y) + C$

Step 14

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

$f (x, y) = \frac{x}{y} + \frac{1}{2} x^{2} + 5 \ln (| y |) - \cos (y) + C$

Step 15

Simplify each term.

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Step 15.1

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

$f (x, y) = \frac{x}{y} + \frac{x^{2}}{2} + 5 \ln (| y |) - \cos (y) + C$

Step 15.2

Simplify $5 \ln (| y |)$ by moving $5$ inside the logarithm.

$f (x, y) = \frac{x}{y} + \frac{x^{2}}{2} + \ln ({| y |}^{5}) - \cos (y) + C$