Calculus Examples

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

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 1$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $\frac{1}{y}$ by $1$ .

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

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

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

$\frac{\partial N}{\partial x} = \frac{1}{y} + 0$

Step 2.4.2

Add $\frac{1}{y}$ and $0$ .

$\frac{\partial N}{\partial x} = \frac{1}{y}$

Step 3

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

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Step 3.1

Substitute $0$ for $\frac{\partial M}{\partial y}$ and $\frac{1}{y}$ for $\frac{\partial N}{\partial x}$ .

$0 = \frac{1}{y}$

Step 3.2

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

$0 = \frac{1}{y}$ 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 $\frac{1}{y}$ for $\frac{\partial N}{\partial x}$ .

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

Step 4.2

Substitute $0$ for $\frac{\partial M}{\partial y}$ .

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

Step 4.3

Substitute $1$ for $M$ .

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Step 4.3.1

Substitute $1$ for $M$ .

$\frac{\frac{1}{y} + 0}{1}$

Step 4.3.2

Divide $\frac{1}{y} + 0$ by $1$ .

$\frac{1}{y} + 0$

Step 4.3.3

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

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Step 5.1

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

$μ (x, y) = e^{\ln (| y |) + C}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Simplify.

$μ (x, y) = e^{\ln (| y |)} + C$

Step 5.2.2

Exponentiation and log are inverse functions.

$μ (x, y) = | y | + C$

Step 6

Multiply $1$ by $y$ .

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

Step 7

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

$f (x, y) = \int y d x$

Step 8

Integrate $M (x, y) = y$ to find $f (x, y)$ .

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Step 8.1

Apply the constant rule.

$f (x, y) = y x + C$

Step 9

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

$f (x, y) = y x + g (y)$

Step 10

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

$\frac{\partial f}{\partial y} = x - y \sin (y)$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

Multiply $x$ by $1$ .

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

Subtract $x$ from both sides of the equation.

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

Step 12.1.2

Combine the opposite terms in $x - y \sin (y) - x$ .

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Step 12.1.2.1

Subtract $x$ from $x$ .

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

Step 12.1.2.2

Add $- y \sin (y)$ and $0$ .

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = y$ and $d v = \sin (y)$ .

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

Step 13.5

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

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

Step 13.6

Simplify.

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Step 13.6.1

Multiply $- 1$ by $- 1$ .

$g (y) = - (y (- \cos (y)) + 1 \int \cos (y) d y)$

Step 13.6.2

Multiply $\int \cos (y) d y$ by $1$ .

$g (y) = - (y (- \cos (y)) + \int \cos (y) d y)$

Step 13.7

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

$g (y) = - (y (- \cos (y)) + \sin (y) + C)$

Step 13.8

Rewrite $- (y (- \cos (y)) + \sin (y) + C)$ as $- (- y \cos (y) + \sin (y)) + C$ .

$g (y) = - (- y \cos (y) + \sin (y)) + C$

Step 14

Substitute for $g (y)$ in $f (x, y) = y x + g (y)$ .

$f (x, y) = y x - (- y \cos (y) + \sin (y)) + C$

Step 15

Simplify each term.

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Step 15.1

Apply the distributive property.

$f (x, y) = y x - (- y \cos (y)) - \sin (y) + C$

Step 15.2

Multiply $- (- y \cos (y))$ .

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Step 15.2.1

Multiply $- 1$ by $- 1$ .

$f (x, y) = y x + 1 (y \cos (y)) - \sin (y) + C$

Step 15.2.2

Multiply $y$ by $1$ .

$f (x, y) = y x + y \cos (y) - \sin (y) + C$