Calculus Examples

$x d y = (x \sin (x) - y) d x$

Step 1

Rewrite the differential equation to fit the Exact differential equation technique.

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Step 1.1

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

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

Step 1.2

Rewrite.

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

Step 2

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

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Step 2.1

Differentiate $M$ with respect to $y$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $0$ and $\frac{d}{d y} [- y]$ .

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

Step 2.6

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

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

Step 2.7

Multiply.

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Step 2.7.1

Multiply $- 1$ by $- 1$ .

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

Step 2.7.2

Multiply $\frac{d}{d y} [y]$ by $1$ .

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

Step 2.8

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

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

Step 3

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

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Step 3.1

Differentiate $N$ with respect to $x$ .

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

Step 3.2

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

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

Step 4

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

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Step 4.1

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

$1 = 1$

Step 4.2

Since the two sides have been shown to be equivalent, the equation is an identity.

$1 = 1$ is an identity.

Step 5

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

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

Step 6

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

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Step 6.1

Apply the constant rule.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

Differentiate $f$ with respect to $x$ .

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

Step 9.2

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

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

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

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

Step 9.3.3

Multiply $y$ by $1$ .

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

Step 9.4

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

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

Step 9.5

Reorder terms.

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

Step 10

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

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Step 10.1

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

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Step 10.1.1

Simplify the right side.

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Step 10.1.1.1

Simplify $- (x \sin (x) - y)$ .

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Step 10.1.1.1.1

Rewrite.

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

Step 10.1.1.1.2

Simplify by adding zeros.

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

Step 10.1.1.1.3

Apply the distributive property.

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

Step 10.1.1.1.4

Multiply $- - y$ .

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Step 10.1.1.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 10.1.1.1.4.2

Multiply $y$ by $1$ .

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

Step 10.1.2

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

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Step 10.1.2.1

Subtract $y$ from both sides of the equation.

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

Step 10.1.2.2

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

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Step 10.1.2.2.1

Subtract $y$ from $y$ .

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

Step 10.1.2.2.2

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

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

Step 11

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

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Step 11.1

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 11.2

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

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

Step 11.3

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

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

Step 11.4

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 11.5

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 11.6

Simplify.

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Step 11.6.1

Multiply $- 1$ by $- 1$ .

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

Step 11.6.2

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

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

Step 11.7

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

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

Step 11.8

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

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

Step 12

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

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

Step 13

Simplify each term.

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Step 13.1

Apply the distributive property.

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

Step 13.2

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

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Step 13.2.1

Multiply $- 1$ by $- 1$ .

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

Step 13.2.2

Multiply $x$ by $1$ .

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