Calculus Examples

$d x + (x - y + 6) d y = 0$

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 $N (x, y) = x - y + 6$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x - y + 6]$

Step 2.2

By the Sum Rule, the derivative of $x - y + 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- y] + \frac{d}{d x} [6]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x] + \frac{d}{d x} [- y] + \frac{d}{d x} [6]$

Step 2.3

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

$\frac{\partial N}{\partial x} = 1 + \frac{d}{d x} [- y] + \frac{d}{d x} [6]$

Step 2.4

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

$\frac{\partial N}{\partial x} = 1 + 0 + \frac{d}{d x} [6]$

Step 2.5

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

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

Step 2.6

Combine terms.

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Step 2.6.1

Add $1$ and $0$ .

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

Step 2.6.2

Add $1$ and $0$ .

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

Step 3

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

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Step 3.1

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

$0 = 1$

Step 3.2

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

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

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

Step 4.2

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

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

Step 4.3

Substitute $1$ for $M$ .

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Step 4.3.1

Substitute $1$ for $M$ .

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

Step 4.3.2

Divide $1 + 0$ by $1$ .

$1 + 0$

Step 4.3.3

Substitute $1$ for $M$ .

$1$

Step 4.4

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

$μ (x, y) = e^{\int d y}$

Step 5

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

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Step 5.1

Apply the constant rule.

$μ (x, y) = e^{y + C}$

Step 5.2

Simplify.

$μ (x, y) = e^{y} + C$

Step 6

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

$d x + (x - y + 6) d y = 0$

Step 7

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

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

Step 8

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

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Step 8.1

Apply the constant rule.

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

Step 9

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

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

Step 10

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

$\frac{\partial f}{\partial y} = x e^{y} - y e^{y} + 6 e^{y}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [e^{y} x + g (y)] = x e^{y} - y e^{y} + 6 e^{y}$

Step 11.2

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

$\frac{d}{d y} [e^{y} x] + \frac{d}{d y} [g (y)] = x e^{y} - y e^{y} + 6 e^{y}$

Step 11.3

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

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Step 11.3.1

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

$x \frac{d}{d y} [e^{y}] + \frac{d}{d y} [g (y)] = x e^{y} - y e^{y} + 6 e^{y}$

Step 11.3.2

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

$x e^{y} + \frac{d}{d y} [g (y)] = x e^{y} - y e^{y} + 6 e^{y}$

Step 11.4

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

$x e^{y} + \frac{d g}{d y} = x e^{y} - y e^{y} + 6 e^{y}$

Step 11.5

Simplify.

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Step 11.5.1

Reorder terms.

$\frac{d g}{d y} + e^{y} x = x e^{y} - y e^{y} + 6 e^{y}$

Step 11.5.2

Reorder factors in $\frac{d g}{d y} + e^{y} x$ .

$\frac{d g}{d y} + x e^{y} = x e^{y} - y e^{y} + 6 e^{y}$

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 $x e^{y}$ from both sides of the equation.

$\frac{d g}{d y} = x e^{y} - y e^{y} + 6 e^{y} - x e^{y}$

Step 12.1.2

Combine the opposite terms in $x e^{y} - y e^{y} + 6 e^{y} - x e^{y}$ .

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Step 12.1.2.1

Subtract $x e^{y}$ from $x e^{y}$ .

$\frac{d g}{d y} = - y e^{y} + 6 e^{y} + 0$

Step 12.1.2.2

Add $- y e^{y} + 6 e^{y}$ and $0$ .

$\frac{d g}{d y} = - y e^{y} + 6 e^{y}$

Step 13

Find the antiderivative of $- y e^{y} + 6 e^{y}$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = - y e^{y} + 6 e^{y}$ .

$\int \frac{d g}{d y} d y = \int - y e^{y} + 6 e^{y} d y$

Step 13.2

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

$g (y) = \int - y e^{y} + 6 e^{y} d y$

Step 13.3

Split the single integral into multiple integrals.

$g (y) = \int - y e^{y} d y + \int 6 e^{y} d y$

Step 13.4

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

$g (y) = - \int y e^{y} d y + \int 6 e^{y} d y$

Step 13.5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = y$ and $d v = e^{y}$ .

$g (y) = - (y e^{y} - \int e^{y} d y) + \int 6 e^{y} d y$

Step 13.6

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

$g (y) = - (y e^{y} - (e^{y} + C)) + \int 6 e^{y} d y$

Step 13.7

Since $6$ is constant with respect to $y$ , move $6$ out of the integral.

$g (y) = - (y e^{y} - (e^{y} + C)) + 6 \int e^{y} d y$

Step 13.8

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

$g (y) = - (y e^{y} - (e^{y} + C)) + 6 (e^{y} + C)$

Step 13.9

Simplify.

$g (y) = - y e^{y} + e^{y} + 6 e^{y} + C$

Step 13.10

Add $e^{y}$ and $6 e^{y}$ .

$g (y) = - y e^{y} + 7 e^{y} + C$

Step 14

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

$f (x, y) = e^{y} x - y e^{y} + 7 e^{y} + C$

Step 15

Reorder factors in $f (x, y) = e^{y} x - y e^{y} + 7 e^{y} + C$ .

$f (x, y) = x e^{y} - y e^{y} + 7 e^{y} + C$