Calculus Examples

$(y - x + x y \cot (x)) d x + x d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

$\frac{\partial M}{\partial y} = 1 + 0 + \frac{d}{d y} [x y \cot (x)]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 1 + 0 + x \cot (x) \frac{d}{d y} [y]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 1 + 0 + x \cot (x) \cdot 1$

Step 1.3.3

Multiply $x$ by $1$ .

$\frac{\partial M}{\partial y} = 1 + 0 + x \cot (x)$

Step 1.4

Simplify.

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Step 1.4.1

Add $1$ and $0$ .

$\frac{\partial M}{\partial y} = 1 + x \cot (x)$

Step 1.4.2

Reorder terms.

$\frac{\partial M}{\partial y} = x \cot (x) + 1$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 3

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

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Step 3.1

Substitute $x \cot (x) + 1$ for $\frac{\partial M}{\partial y}$ and $1$ for $\frac{\partial N}{\partial x}$ .

$x \cot (x) + 1 = 1$

Step 3.2

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

$x \cot (x) + 1 = 1$ 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 $x \cot (x) + 1$ for $\frac{\partial M}{\partial y}$ .

$\frac{x \cot (x) + 1 - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $1$ for $\frac{\partial N}{\partial x}$ .

$\frac{x \cot (x) + 1 - 1}{N}$

Step 4.3

Substitute $x$ for $N$ .

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Step 4.3.1

Substitute $x$ for $N$ .

$\frac{x \cot (x) + 1 - 1}{x}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Subtract $1$ from $1$ .

$\frac{x \cot (x) + 0}{x}$

Step 4.3.2.2

Add $x \cot (x)$ and $0$ .

$\frac{x \cot (x)}{x}$

Step 4.3.3

Cancel the common factor of $x$ .

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Step 4.3.3.1

Cancel the common factor.

$\frac{x \cot (x)}{x}$

Step 4.3.3.2

Divide $\cot (x)$ by $1$ .

$\cot (x)$

Step 4.4

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

$μ (x, y) = e^{\int \cot (x) d x}$

Step 5

Evaluate the integral $e^{\int \cot (x) d x}$ .

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Step 5.1

The integral of $\cot (x)$ with respect to $x$ is $\ln (| \sin (x) |)$ .

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

Step 5.2

Simplify the answer.

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Step 5.2.1

Simplify.

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

Step 5.2.2

Exponentiation and log are inverse functions.

$μ (x, y) = | \sin (x) | + C$

Step 6

Multiply both sides of $(y - x + x y \cot (x)) d x + x d y = 0$ by the integration factor $\sin (x)$ .

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Step 6.1

Multiply $y - x + x y \cot (x)$ by $\sin (x)$ .

$(y - x + x y \cot (x)) \sin (x) d x + x d y = 0$

Step 6.2

Simplify each term.

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Step 6.2.1

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 6.2.2

Multiply $x y \frac{\cos (x)}{\sin (x)}$ .

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Step 6.2.2.1

Combine $x$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 6.2.2.2

Combine $y$ and $\frac{x \cos (x)}{\sin (x)}$ .

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

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Cancel the common factor of $\sin (x)$ .

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Step 6.4.1

Cancel the common factor.

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

Step 6.4.2

Rewrite the expression.

$(y \sin (x) - x \sin (x) + y x \cos (x)) d x + x d y = 0$

Step 6.5

Multiply $x$ by $\sin (x)$ .

$(y \sin (x) - x \sin (x) + y x \cos (x)) d x + x \sin (x) d y = 0$

Step 7

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

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

Step 8

Integrate $N (x, y) = x \sin (x)$ to find $f (x, y)$ .

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Step 8.1

Apply the constant rule.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \sin (x)$ .

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

Step 11.3.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 11.3.4

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

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

Step 11.3.5

Multiply $\sin (x)$ by $1$ .

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

Step 11.4

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

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

Step 11.5

Simplify.

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Step 11.5.1

Apply the distributive property.

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

Step 11.5.2

Reorder terms.

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.1.3

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

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Step 12.1.3.1

Reorder the factors in the terms $y x \cos (x)$ and $- x y \cos (x)$ .

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

Step 12.1.3.2

Subtract $x y \cos (x)$ from $x y \cos (x)$ .

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

Step 12.1.3.3

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

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

Step 12.1.3.4

Subtract $y \sin (x)$ from $y \sin (x)$ .

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

Step 12.1.3.5

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

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

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

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

Step 13.5

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

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

Step 13.6

Simplify.

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Step 13.6.1

Multiply $- 1$ by $- 1$ .

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

Step 13.6.2

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

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

Step 13.7

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

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

Step 13.8

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

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

Step 14

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

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

Step 15

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

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Step 15.1

Simplify each term.

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Step 15.1.1

Apply the distributive property.

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

Step 15.1.2

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

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Step 15.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 15.1.2.2

Multiply $x$ by $1$ .

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

Step 15.2

Reorder factors in $f (x, y) = x \sin (x) y + x \cos (x) - \sin (x) + C$ .

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