Calculus Examples

$y d x + (3 + 3 x - y) d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [3 + 3 x - y]$

Step 2.2

Differentiate.

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Step 2.2.1

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

$\frac{\partial N}{\partial x} = \frac{d}{d x} [3] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- y]$

Step 2.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $3$ by $1$ .

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

Step 2.4

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

$\frac{\partial N}{\partial x} = 0 + 3 + 0$

Step 2.5

Combine terms.

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Step 2.5.1

Add $0$ and $3$ .

$\frac{\partial N}{\partial x} = 3 + 0$

Step 2.5.2

Add $3$ and $0$ .

$\frac{\partial N}{\partial x} = 3$

Step 3

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

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Step 3.1

Substitute $1$ for $\frac{\partial M}{\partial y}$ and $3$ for $\frac{\partial N}{\partial x}$ .

$1 = 3$

Step 3.2

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

$1 = 3$ 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 $3$ for $\frac{\partial N}{\partial x}$ .

$\frac{3 - \frac{\partial M}{\partial y}}{M}$

Step 4.2

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

$\frac{3 - 1}{M}$

Step 4.3

Substitute $y$ for $M$ .

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Step 4.3.1

Substitute $y$ for $M$ .

$\frac{3 - 1}{y}$

Step 4.3.2

Substitute $y$ for $M$ .

$\frac{2}{y}$

Step 4.4

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

$μ (x, y) = e^{\int \frac{2}{y} d y}$

Step 5

Evaluate the integral $e^{\int \frac{2}{y} d y}$ .

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Step 5.1

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$μ (x, y) = e^{2 \int \frac{1}{y} d y}$

Step 5.2

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

$μ (x, y) = e^{2 (\ln (| y |) + C)}$

Step 5.3

Simplify.

$μ (x, y) = e^{2 \ln (| y |)} + C$

Step 5.4

Simplify each term.

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Step 5.4.1

Simplify $2 \ln (| y |)$ by moving $2$ inside the logarithm.

$μ (x, y) = e^{\ln ({| y |}^{2})} + C$

Step 5.4.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| y |}^{2} + C$

Step 5.4.3

Remove the absolute value in ${| y |}^{2}$ because exponentiations with even powers are always positive.

$μ (x, y) = y^{2} + C$

Step 6

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

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Step 6.1

Multiply $y$ by $y^{2}$ .

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

Step 6.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 6.2.1

Multiply $y$ by $y^{2}$ .

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Step 6.2.1.1

Raise $y$ to the power of $1$ .

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

Step 6.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.2.2

Add $1$ and $2$ .

$y^{3} d x + (3 + 3 x - y) d y = 0$

Step 6.3

Multiply $3 + 3 x - y$ by $y^{2}$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 6.5.1

Move $y^{2}$ .

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

Step 6.5.2

Multiply $y^{2}$ by $y$ .

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Step 6.5.2.1

Raise $y$ to the power of $1$ .

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

Step 6.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.5.3

Add $2$ and $1$ .

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

Step 7

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

$f (x, y) = \int y^{3} d x$

Step 8

Integrate $M (x, y) = y^{3}$ to find $f (x, y)$ .

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Step 8.1

Apply the constant rule.

$f (x, y) = y^{3} x + C$

Step 9

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

$f (x, y) = y^{3} x + g (y)$

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [y^{3} x + g (y)] = 3 y^{2} + 3 x y^{2} - y^{3}$

Step 11.2

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

$\frac{d}{d y} [y^{3} x] + \frac{d}{d y} [g (y)] = 3 y^{2} + 3 x y^{2} - y^{3}$

Step 11.3

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

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Step 11.3.1

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

$x \frac{d}{d y} [y^{3}] + \frac{d}{d y} [g (y)] = 3 y^{2} + 3 x y^{2} - y^{3}$

Step 11.3.2

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

$x (3 y^{2}) + \frac{d}{d y} [g (y)] = 3 y^{2} + 3 x y^{2} - y^{3}$

Step 11.3.3

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

$3 x y^{2} + \frac{d}{d y} [g (y)] = 3 y^{2} + 3 x y^{2} - y^{3}$

Step 11.4

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

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

Step 11.5

Reorder terms.

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

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 $3 y^{2} x$ from both sides of the equation.

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

Step 12.1.2

Combine the opposite terms in $3 y^{2} + 3 x y^{2} - y^{3} - 3 y^{2} x$ .

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Step 12.1.2.1

Reorder the factors in the terms $3 x y^{2}$ and $- 3 y^{2} x$ .

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

Step 12.1.2.2

Subtract $3 y^{2} x$ from $3 y^{2} x$ .

$\frac{d g}{d y} = 3 y^{2} - y^{3} + 0$

Step 12.1.2.3

Add $3 y^{2} - y^{3}$ and $0$ .

$\frac{d g}{d y} = 3 y^{2} - y^{3}$

Step 13

Find the antiderivative of $3 y^{2} - y^{3}$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = 3 y^{2} - y^{3}$ .

$\int \frac{d g}{d y} d y = \int 3 y^{2} - y^{3} d y$

Step 13.2

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

$g (y) = \int 3 y^{2} - y^{3} d y$

Step 13.3

Split the single integral into multiple integrals.

$g (y) = \int 3 y^{2} d y + \int - y^{3} d y$

Step 13.4

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

$g (y) = 3 \int y^{2} d y + \int - y^{3} d y$

Step 13.5

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

$g (y) = 3 (\frac{1}{3} y^{3} + C) + \int - y^{3} d y$

Step 13.6

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

$g (y) = 3 (\frac{1}{3} y^{3} + C) - \int y^{3} d y$

Step 13.7

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

$g (y) = 3 (\frac{1}{3} y^{3} + C) - (\frac{1}{4} y^{4} + C)$

Step 13.8

Simplify.

$g (y) = 3 (\frac{1}{3}) y^{3} - \frac{1}{4} y^{4} + C$

Step 13.9

Simplify.

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Step 13.9.1

Combine $3$ and $\frac{1}{3}$ .

$g (y) = \frac{3}{3} y^{3} - \frac{1}{4} y^{4} + C$

Step 13.9.2

Cancel the common factor of $3$ .

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Step 13.9.2.1

Cancel the common factor.

$g (y) = \frac{3}{3} y^{3} - \frac{1}{4} y^{4} + C$

Step 13.9.2.2

Rewrite the expression.

$g (y) = 1 y^{3} - \frac{1}{4} y^{4} + C$

Step 13.9.3

Multiply $y^{3}$ by $1$ .

$g (y) = y^{3} - \frac{1}{4} y^{4} + C$

Step 14

Substitute for $g (y)$ in $f (x, y) = y^{3} x + g (y)$ .

$f (x, y) = y^{3} x + y^{3} - \frac{1}{4} y^{4} + C$

Step 15

Combine $y^{4}$ and $\frac{1}{4}$ .

$f (x, y) = y^{3} x + y^{3} - \frac{y^{4}}{4} + C$