Calculus Examples

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

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = x^{2} + y^{2} + 1$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

By the Sum Rule, the derivative of $x^{2} + y^{2} + 1$ with respect to $y$ is $\frac{d}{d y} [x^{2}] + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [1]$ .

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

Step 1.3

Since $x^{2}$ is constant with respect to $y$ , the derivative of $x^{2}$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [1]$

Step 1.4

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

$\frac{\partial M}{\partial y} = 0 + 2 y + \frac{d}{d y} [1]$

Step 1.5

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 0 + 2 y + 0$

Step 1.6

Combine terms.

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Step 1.6.1

Add $0$ and $2 y$ .

$\frac{\partial M}{\partial y} = 2 y + 0$

Step 1.6.2

Add $2 y$ and $0$ .

$\frac{\partial M}{\partial y} = 2 y$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $- 2$ by $1$ .

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

Step 3

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

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Step 3.1

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

$2 y = 2 x - 2 y$

Step 3.2

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

$2 y = 2 x - 2 y$ 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 $2 y$ for $\frac{\partial M}{\partial y}$ .

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

Step 4.2

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

$\frac{2 y - (2 x - 2 y)}{N}$

Step 4.3

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

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Step 4.3.1

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

$\frac{2 y - (2 x - 2 y)}{x^{2} - 2 x y}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

$\frac{2 y - (2 x) - (- 2 y)}{x^{2} - 2 x y}$

Step 4.3.2.2

Multiply $2$ by $- 1$ .

$\frac{2 y - 2 x - (- 2 y)}{x^{2} - 2 x y}$

Step 4.3.2.3

Multiply $- 2$ by $- 1$ .

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

Step 4.3.2.4

Add $2 y$ and $2 y$ .

$\frac{- 2 x + 4 y}{x^{2} - 2 x y}$

Step 4.3.2.5

Factor $2$ out of $- 2 x + 4 y$ .

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Step 4.3.2.5.1

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

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

Step 4.3.2.5.2

Factor $2$ out of $4 y$ .

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

Step 4.3.2.5.3

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

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

Step 4.3.3

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

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Step 4.3.3.1

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

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

Step 4.3.3.2

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

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

Step 4.3.3.3

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

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

Step 4.3.4

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

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Step 4.3.4.1

Factor $- 1$ out of $- x$ .

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

Step 4.3.4.2

Factor $- 1$ out of $2 y$ .

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

Step 4.3.4.3

Factor $- 1$ out of $- (x) - (- 2 y)$ .

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

Step 4.3.4.4

Rewrite $- (x - 2 y)$ as $- 1 (x - 2 y)$ .

$\frac{2 (- 1 (x - 2 y))}{x (x - 2 y)}$

Step 4.3.4.5

Cancel the common factor.

$\frac{2 (- 1 (x - 2 y))}{x (x - 2 y)}$

Step 4.3.4.6

Rewrite the expression.

$\frac{2 \cdot (- 1)}{x}$

Step 4.3.5

Multiply $2$ by $- 1$ .

$\frac{- 2}{x}$

Step 4.3.6

Move the negative in front of the fraction.

$- \frac{2}{x}$

Step 4.4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Multiply $2$ by $- 1$ .

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

Step 5.4

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

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

Step 5.5

Simplify.

$μ (x, y) = e^{- 2 \ln (| x |)} + C$

Step 5.6

Simplify each term.

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Step 5.6.1

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

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

Step 5.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| x |}^{- 2} + C$

Step 5.6.3

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

$μ (x, y) = x^{- 2} + C$

Step 5.6.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$μ (x, y) = \frac{1}{x^{2}} + C$

Step 6

Multiply both sides of $(x^{2} + y^{2} + 1) d x + (x^{2} - 2 x y) d y = 0$ by the integration factor $\frac{1}{x^{2}}$ .

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Step 6.1

Multiply $x^{2} + y^{2} + 1$ by $\frac{1}{x^{2}}$ .

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

Step 6.2

Multiply $x^{2} + y^{2} + 1$ by $\frac{1}{x^{2}}$ .

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

Step 6.3

Multiply $x^{2} - 2 x y$ by $\frac{1}{x^{2}}$ .

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

Step 6.4

Multiply $x^{2} - 2 x y$ by $\frac{1}{x^{2}}$ .

$\frac{x^{2} + y^{2} + 1}{x^{2}} d x + \frac{x^{2} - 2 x y}{x^{2}} d y = 0$

Step 6.5

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

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Step 6.5.1

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

$\frac{x^{2} + y^{2} + 1}{x^{2}} d x + \frac{x \cdot x - 2 x y}{x^{2}} d y = 0$

Step 6.5.2

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

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

Step 6.5.3

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

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

Step 6.6

Cancel the common factors.

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Step 6.6.1

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

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

Step 6.6.2

Cancel the common factor.

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

Step 6.6.3

Rewrite the expression.

$\frac{x^{2} + y^{2} + 1}{x^{2}} d x + \frac{x - 2 y}{x} d y = 0$

Step 7

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

$f (x, y) = \int \frac{x - 2 y}{x} d y$

Step 8

Integrate $N (x, y) = \frac{x - 2 y}{x}$ to find $f (x, y)$ .

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Step 8.1

Split the fraction into multiple fractions.

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

Step 8.2

Split the single integral into multiple integrals.

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

Step 8.3

Simplify.

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Step 8.3.1

Cancel the common factor of $x$ .

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Step 8.3.1.1

Cancel the common factor.

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

Step 8.3.1.2

Rewrite the expression.

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

Step 8.3.2

Move the negative in front of the fraction.

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

Step 8.4

Apply the constant rule.

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

Step 8.5

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

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

Step 8.6

Since $\frac{2}{x}$ is constant with respect to $y$ , move $\frac{2}{x}$ out of the integral.

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

Step 8.7

Remove parentheses.

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

Step 8.8

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

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

Step 8.9

Simplify.

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

Step 8.10

Simplify.

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Step 8.10.1

Multiply $\frac{1}{2}$ by $\frac{2}{x}$ .

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

Step 8.10.2

Cancel the common factor of $2$ .

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Step 8.10.2.1

Cancel the common factor.

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

Step 8.10.2.2

Rewrite the expression.

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

Step 8.10.3

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

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

Step 9

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

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

Step 10

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

$\frac{\partial f}{\partial x} = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

Differentiate.

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Step 11.2.1

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

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

Step 11.2.2

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

$0 + \frac{d}{d x} [- \frac{y^{2}}{x}] + \frac{d}{d x} [g (x)] = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11.3

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

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Step 11.3.1

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

$0 - y^{2} \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [g (x)] = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$0 - y^{2} \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [g (x)] = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11.3.3

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

$0 - y^{2} (- x^{- 2}) + \frac{d}{d x} [g (x)] = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11.3.4

Multiply $- 1$ by $- 1$ .

$0 + 1 y^{2} x^{- 2} + \frac{d}{d x} [g (x)] = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11.3.5

Multiply $y^{2}$ by $1$ .

$0 + y^{2} x^{- 2} + \frac{d}{d x} [g (x)] = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11.4

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

$0 + y^{2} x^{- 2} + \frac{d g}{d x} = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11.5

Simplify.

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Step 11.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + y^{2} \frac{1}{x^{2}} + \frac{d g}{d x} = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11.5.2

Combine terms.

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Step 11.5.2.1

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

$0 + \frac{y^{2}}{x^{2}} + \frac{d g}{d x} = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11.5.2.2

Add $0$ and $\frac{y^{2}}{x^{2}}$ .

$\frac{y^{2}}{x^{2}} + \frac{d g}{d x} = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 11.5.3

Reorder terms.

$\frac{d g}{d x} + \frac{y^{2}}{x^{2}} = \frac{x^{2} + y^{2} + 1}{x^{2}}$

Step 12

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

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Step 12.1

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

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Step 12.1.1

Move all terms containing variables to the left side of the equation.

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Step 12.1.1.1

Subtract $\frac{x^{2} + y^{2} + 1}{x^{2}}$ from both sides of the equation.

$\frac{d g}{d x} + \frac{y^{2}}{x^{2}} - \frac{x^{2} + y^{2} + 1}{x^{2}} = 0$

Step 12.1.1.2

Combine the numerators over the common denominator.

$\frac{d g}{d x} + \frac{y^{2} - (x^{2} + y^{2} + 1)}{x^{2}} = 0$

Step 12.1.1.3

Simplify each term.

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Step 12.1.1.3.1

Apply the distributive property.

$\frac{d g}{d x} + \frac{y^{2} - x^{2} - y^{2} - 1 \cdot 1}{x^{2}} = 0$

Step 12.1.1.3.2

Multiply $- 1$ by $1$ .

$\frac{d g}{d x} + \frac{y^{2} - x^{2} - y^{2} - 1}{x^{2}} = 0$

Step 12.1.1.4

Combine the opposite terms in $y^{2} - x^{2} - y^{2} - 1$ .

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Step 12.1.1.4.1

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

$\frac{d g}{d x} + \frac{- x^{2} + 0 - 1}{x^{2}} = 0$

Step 12.1.1.4.2

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

$\frac{d g}{d x} + \frac{- x^{2} - 1}{x^{2}} = 0$

Step 12.1.1.5

To write $\frac{d g}{d x}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{\frac{d g}{d x} x^{2}}{x^{2}} + \frac{- x^{2} - 1}{x^{2}} = 0$

Step 12.1.1.6

Combine the numerators over the common denominator.

$\frac{\frac{d g}{d x} x^{2} - x^{2} - 1}{x^{2}} = 0$

Step 12.1.2

Set the numerator equal to zero.

$\frac{d g}{d x} x^{2} - x^{2} - 1 = 0$

Step 12.1.3

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

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Step 12.1.3.1

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

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Step 12.1.3.1.1

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

$\frac{d g}{d x} x^{2} - 1 = x^{2}$

Step 12.1.3.1.2

Add $1$ to both sides of the equation.

$\frac{d g}{d x} x^{2} = x^{2} + 1$

Step 12.1.3.2

Divide each term in $\frac{d g}{d x} x^{2} = x^{2} + 1$ by $x^{2}$ and simplify.

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Step 12.1.3.2.1

Divide each term in $\frac{d g}{d x} x^{2} = x^{2} + 1$ by $x^{2}$ .

$\frac{\frac{d g}{d x} x^{2}}{x^{2}} = \frac{x^{2}}{x^{2}} + \frac{1}{x^{2}}$

Step 12.1.3.2.2

Simplify the left side.

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Step 12.1.3.2.2.1

Cancel the common factor of $x^{2}$ .

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Step 12.1.3.2.2.1.1

Cancel the common factor.

$\frac{\frac{d g}{d x} x^{2}}{x^{2}} = \frac{x^{2}}{x^{2}} + \frac{1}{x^{2}}$

Step 12.1.3.2.2.1.2

Divide $\frac{d g}{d x}$ by $1$ .

$\frac{d g}{d x} = \frac{x^{2}}{x^{2}} + \frac{1}{x^{2}}$

Step 12.1.3.2.3

Simplify the right side.

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Step 12.1.3.2.3.1

Cancel the common factor of $x^{2}$ .

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Step 12.1.3.2.3.1.1

Cancel the common factor.

$\frac{d g}{d x} = \frac{x^{2}}{x^{2}} + \frac{1}{x^{2}}$

Step 12.1.3.2.3.1.2

Rewrite the expression.

$\frac{d g}{d x} = 1 + \frac{1}{x^{2}}$

Step 13

Find the antiderivative of $1 + \frac{1}{x^{2}}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = 1 + \frac{1}{x^{2}}$ .

$\int \frac{d g}{d x} d x = \int 1 + \frac{1}{x^{2}} d x$

Step 13.2

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

$g (x) = \int 1 + \frac{1}{x^{2}} d x$

Step 13.3

Split the single integral into multiple integrals.

$g (x) = \int d x + \int \frac{1}{x^{2}} d x$

Step 13.4

Apply the constant rule.

$g (x) = x + C + \int \frac{1}{x^{2}} d x$

Step 13.5

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$g (x) = x + C + \int {(x^{2})}^{- 1} d x$

Step 13.6

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 13.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$g (x) = x + C + \int x^{2 \cdot - 1} d x$

Step 13.6.2

Multiply $2$ by $- 1$ .

$g (x) = x + C + \int x^{- 2} d x$

Step 13.7

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

$g (x) = x + C - x^{- 1} + C$

Step 13.8

Simplify.

$g (x) = x - \frac{1}{x} + C$

Step 14

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

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