Calculus Examples

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

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 3 x y - 2 a y^{2}$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [3 x y - 2 a y^{2}]$

Step 1.2

By the Sum Rule, the derivative of $3 x y - 2 a y^{2}$ with respect to $y$ is $\frac{d}{d y} [3 x y] + \frac{d}{d y} [- 2 a y^{2}]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [3 x y] + \frac{d}{d y} [- 2 a y^{2}]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 3 x \frac{d}{d y} [y] + \frac{d}{d y} [- 2 a y^{2}]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 3 x \cdot 1 + \frac{d}{d y} [- 2 a y^{2}]$

Step 1.3.3

Multiply $3$ by $1$ .

$\frac{\partial M}{\partial y} = 3 x + \frac{d}{d y} [- 2 a y^{2}]$

Step 1.4

Evaluate $\frac{d}{d y} [- 2 a y^{2}]$ .

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Step 1.4.1

Since $- 2 a$ is constant with respect to $y$ , the derivative of $- 2 a y^{2}$ with respect to $y$ is $- 2 a \frac{d}{d y} [y^{2}]$ .

$\frac{\partial M}{\partial y} = 3 x - 2 a \frac{d}{d y} [y^{2}]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$\frac{\partial M}{\partial y} = 3 x - 2 a (2 y)$

Step 1.4.3

Multiply $2$ by $- 2$ .

$\frac{\partial M}{\partial y} = 3 x - 4 a y$

Step 1.5

Reorder terms.

$\frac{\partial M}{\partial y} = - 4 a y + 3 x$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = x^{2} - 2 a y x$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2} - 2 a y x]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $x^{2} - 2 a y x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 a y x]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 a y x]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{\partial N}{\partial x} = 2 x + \frac{d}{d x} [- 2 a y x]$

Step 2.3

Evaluate $\frac{d}{d x} [- 2 a y x]$ .

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Step 2.3.1

Since $- 2 a y$ is constant with respect to $x$ , the derivative of $- 2 a y x$ with respect to $x$ is $- 2 a y \frac{d}{d x} [x]$ .

$\frac{\partial N}{\partial x} = 2 x - 2 a y \frac{d}{d x} [x]$

Step 2.3.2

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

$\frac{\partial N}{\partial x} = 2 x - 2 a y \cdot 1$

Step 2.3.3

Multiply $- 2$ by $1$ .

$\frac{\partial N}{\partial x} = 2 x - 2 a y$

Step 2.4

Reorder terms.

$\frac{\partial N}{\partial x} = - 2 a y + 2 x$

Step 3

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

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Step 3.1

Substitute $- 4 a y + 3 x$ for $\frac{\partial M}{\partial y}$ and $- 2 a y + 2 x$ for $\frac{\partial N}{\partial x}$ .

$- 4 a y + 3 x = - 2 a y + 2 x$

Step 3.2

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

$- 4 a y + 3 x = - 2 a y + 2 x$ 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 $- 4 a y + 3 x$ for $\frac{\partial M}{\partial y}$ .

$\frac{- 4 a y + 3 x - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $- 2 a y + 2 x$ for $\frac{\partial N}{\partial x}$ .

$\frac{- 4 a y + 3 x - (- 2 a y + 2 x)}{N}$

Step 4.3

Substitute $x^{2} - 2 a y x$ for $N$ .

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Step 4.3.1

Substitute $x^{2} - 2 a y x$ for $N$ .

$\frac{- 4 a y + 3 x - (- 2 a y + 2 x)}{x^{2} - 2 a y x}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

$\frac{- 4 a y + 3 x - (- 2 a y) - (2 x)}{x^{2} - 2 a y x}$

Step 4.3.2.2

Multiply $- 2$ by $- 1$ .

$\frac{- 4 a y + 3 x + 2 (a y) - (2 x)}{x^{2} - 2 a y x}$

Step 4.3.2.3

Multiply $2$ by $- 1$ .

$\frac{- 4 a y + 3 x + 2 (a y) - 2 x}{x^{2} - 2 a y x}$

Step 4.3.2.4

Add $- 4 a y$ and $2 a y$ .

$\frac{- 2 a y + 3 x - 2 x}{x^{2} - 2 a y x}$

Step 4.3.2.5

Subtract $2 x$ from $3 x$ .

$\frac{- 2 a y + x}{x^{2} - 2 a y x}$

Step 4.3.3

Factor $x$ out of $x^{2} - 2 a y x$ .

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Step 4.3.3.1

Factor $x$ out of $x^{2}$ .

$\frac{- 2 a y + x}{x \cdot x - 2 a y x}$

Step 4.3.3.2

Factor $x$ out of $- 2 a y x$ .

$\frac{- 2 a y + x}{x \cdot x + x (- 2 a y)}$

Step 4.3.3.3

Factor $x$ out of $x \cdot x + x (- 2 a y)$ .

$\frac{- 2 a y + x}{x (x - 2 a y)}$

Step 4.3.4

Cancel the common factor of $- 2 a y + x$ and $x - 2 a y$ .

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Step 4.3.4.1

Reorder terms.

$\frac{- 2 a y + x}{x (- 2 a y + x)}$

Step 4.3.4.2

Cancel the common factor.

$\frac{- 2 a y + x}{x (- 2 a y + x)}$

Step 4.3.4.3

Rewrite the expression.

$\frac{1}{x}$

Step 4.4

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

$μ (x, y) = e^{\int \frac{1}{x} d x}$

Step 5

Evaluate the integral $e^{\int \frac{1}{x} d x}$ .

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Step 5.1

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

$μ (x, y) = e^{\ln (| x |) + C}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Simplify.

$μ (x, y) = e^{\ln (| x |)} + C$

Step 5.2.2

Exponentiation and log are inverse functions.

$μ (x, y) = | x | + C$

Step 6

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

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1

Move $x$ .

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

Step 6.3.2

Multiply $x$ by $x$ .

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

Step 6.4

Multiply $x^{2} - 2 a y x$ by $x$ .

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

Step 6.5

Apply the distributive property.

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

Step 6.6

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

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Step 6.6.1

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

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Step 6.6.1.1

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

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

Step 6.6.1.2

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

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

Step 6.6.2

Add $2$ and $1$ .

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

Step 6.7

Multiply $x$ by $x$ by adding the exponents.

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Step 6.7.1

Move $x$ .

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

Step 6.7.2

Multiply $x$ by $x$ .

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

Step 7

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

$f (x, y) = \int 3 x^{2} y - 2 a y^{2} x d x$

Step 8

Integrate $M (x, y) = 3 x^{2} y - 2 a y^{2} x$ to find $f (x, y)$ .

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Step 8.1

Split the single integral into multiple integrals.

$f (x, y) = \int 3 x^{2} y d x + \int - 2 a y^{2} x d x$

Step 8.2

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

$f (x, y) = 3 y \int x^{2} d x + \int - 2 a y^{2} x d x$

Step 8.3

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

$f (x, y) = 3 y (\frac{1}{3} x^{3} + C) + \int - 2 a y^{2} x d x$

Step 8.4

Since $- 2 a y^{2}$ is constant with respect to $x$ , move $- 2 a y^{2}$ out of the integral.

$f (x, y) = 3 y (\frac{1}{3} x^{3} + C) - 2 a y^{2} \int x d x$

Step 8.5

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

$f (x, y) = 3 y (\frac{1}{3} x^{3} + C) - 2 a y^{2} (\frac{1}{2} x^{2} + C)$

Step 8.6

Simplify.

$f (x, y) = 3 \cdot \frac{1}{3} y x^{3} - 2 a y^{2} (\frac{1}{2} x^{2}) + C$

Step 8.7

Simplify.

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Step 8.7.1

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

$f (x, y) = \frac{3}{3} y x^{3} - 2 a y^{2} (\frac{1}{2} x^{2}) + C$

Step 8.7.2

Cancel the common factor of $3$ .

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Step 8.7.2.1

Cancel the common factor.

$f (x, y) = \frac{3}{3} y x^{3} - 2 a y^{2} (\frac{1}{2} x^{2}) + C$

Step 8.7.2.2

Rewrite the expression.

$f (x, y) = 1 y x^{3} - 2 a y^{2} (\frac{1}{2} x^{2}) + C$

Step 8.7.3

Multiply $y$ by $1$ .

$f (x, y) = y x^{3} - 2 a y^{2} (\frac{1}{2} x^{2}) + C$

Step 8.7.4

Combine $\frac{1}{2}$ and $x^{2}$ .

$f (x, y) = y x^{3} - 2 a y^{2} \frac{x^{2}}{2} + C$

Step 8.7.5

Combine $- 2$ and $\frac{x^{2}}{2}$ .

$f (x, y) = y x^{3} + a y^{2} \frac{- 2 x^{2}}{2} + C$

Step 8.7.6

Combine $a$ and $\frac{- 2 x^{2}}{2}$ .

$f (x, y) = y x^{3} + y^{2} \frac{a (- 2 x^{2})}{2} + C$

Step 8.7.7

Combine $y^{2}$ and $\frac{a (- 2 x^{2})}{2}$ .

$f (x, y) = y x^{3} + \frac{y^{2} (a (- 2 x^{2}))}{2} + C$

Step 8.7.8

Remove parentheses.

$f (x, y) = y x^{3} + \frac{y^{2} a (- 2 x^{2})}{2} + C$

Step 8.7.9

Cancel the common factor of $- 2$ and $2$ .

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Step 8.7.9.1

Factor $2$ out of $y^{2} a (- 2 x^{2})$ .

$f (x, y) = y x^{3} + \frac{2 (y^{2} a (- x^{2}))}{2} + C$

Step 8.7.9.2

Cancel the common factors.

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Step 8.7.9.2.1

Factor $2$ out of $2$ .

$f (x, y) = y x^{3} + \frac{2 (y^{2} a (- x^{2}))}{2 (1)} + C$

Step 8.7.9.2.2

Cancel the common factor.

$f (x, y) = y x^{3} + \frac{2 (y^{2} a (- x^{2}))}{2 \cdot 1} + C$

Step 8.7.9.2.3

Rewrite the expression.

$f (x, y) = y x^{3} + \frac{y^{2} a (- x^{2})}{1} + C$

Step 8.7.9.2.4

Divide $y^{2} a (- x^{2})$ by $1$ .

$f (x, y) = y x^{3} + y^{2} a (- x^{2}) + C$

Step 9

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

$f (x, y) = y x^{3} + y^{2} a (- x^{2}) + g (y)$

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

By the Sum Rule, the derivative of $y x^{3} + y^{2} a (- x^{2}) + g (y)$ with respect to $y$ is $\frac{d}{d y} [y x^{3}] + \frac{d}{d y} [y^{2} a (- x^{2})] + \frac{d}{d y} [g (y)]$ .

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

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

$x^{3} \cdot 1 + \frac{d}{d y} [y^{2} a (- x^{2})] + \frac{d}{d y} [g (y)] = x^{3} - 2 a y x^{2}$

Step 11.3.3

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

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

Step 11.4

Evaluate $\frac{d}{d y} [y^{2} a (- x^{2})]$ .

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Step 11.4.1

Since $a \cdot - 1 x^{2}$ is constant with respect to $y$ , the derivative of $y^{2} a (- x^{2})$ with respect to $y$ is $a \cdot - 1 x^{2} \frac{d}{d y} [y^{2}]$ .

$x^{3} + a \cdot - 1 x^{2} \frac{d}{d y} [y^{2}] + \frac{d}{d y} [g (y)] = x^{3} - 2 a y x^{2}$

Step 11.4.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$x^{3} + a \cdot - 1 x^{2} (2 y) + \frac{d}{d y} [g (y)] = x^{3} - 2 a y x^{2}$

Step 11.4.3

Move $- 1$ to the left of $a$ .

$x^{3} - 1 \cdot a x^{2} (2 y) + \frac{d}{d y} [g (y)] = x^{3} - 2 a y x^{2}$

Step 11.4.4

Rewrite $- 1 a$ as $- a$ .

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

Step 11.4.5

Multiply $2$ by $- 1$ .

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

Step 11.5

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

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

Step 11.6

Reorder terms.

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

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

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

Step 12.1.2

Add $2 x^{2} a y$ to both sides of the equation.

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

Step 12.1.3

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

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Step 12.1.3.1

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

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

Step 12.1.3.2

Add $- 2 a y x^{2}$ and $0$ .

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

Step 12.1.3.3

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

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

Step 12.1.3.4

Add $- 2 x^{2} a y$ and $2 x^{2} a y$ .

$\frac{d g}{d y} = 0$

Step 13

Find the antiderivative of $0$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = 0$ .

$\int \frac{d g}{d y} d y = \int 0 d y$

Step 13.2

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

$g (y) = \int 0 d y$

Step 13.3

The integral of $0$ with respect to $y$ is $0$ .

$g (y) = 0 + C$

Step 13.4

Add $0$ and $C$ .

$g (y) = C$

Step 14

Substitute for $g (y)$ in $f (x, y) = y x^{3} + y^{2} a (- x^{2}) + g (y)$ .

$f (x, y) = y x^{3} + y^{2} a (- x^{2}) + C$

Step 15

Rewrite using the commutative property of multiplication.

$f (x, y) = y x^{3} - y^{2} a x^{2} + C$