Calculus Examples

$(4 y + x^{2} y) d x + (y - 1) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 4 y + x^{2} y$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [4 y + x^{2} y]$

Step 1.2

By the Sum Rule, the derivative of $4 y + x^{2} y$ with respect to $y$ is $\frac{d}{d y} [4 y] + \frac{d}{d y} [x^{2} y]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [4 y] + \frac{d}{d y} [x^{2} y]$

Step 1.3

Evaluate $\frac{d}{d y} [4 y]$ .

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Step 1.3.1

Since $4$ is constant with respect to $y$ , the derivative of $4 y$ with respect to $y$ is $4 \frac{d}{d y} [y]$ .

$\frac{\partial M}{\partial y} = 4 \frac{d}{d y} [y] + \frac{d}{d y} [x^{2} y]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 4 \cdot 1 + \frac{d}{d y} [x^{2} y]$

Step 1.3.3

Multiply $4$ by $1$ .

$\frac{\partial M}{\partial y} = 4 + \frac{d}{d y} [x^{2} y]$

Step 1.4

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

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Step 1.4.1

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

$\frac{\partial M}{\partial y} = 4 + x^{2} \frac{d}{d y} [y]$

Step 1.4.2

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

$\frac{\partial M}{\partial y} = 4 + x^{2} \cdot 1$

Step 1.4.3

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

$\frac{\partial M}{\partial y} = 4 + x^{2}$

Step 1.5

Reorder terms.

$\frac{\partial M}{\partial y} = x^{2} + 4$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = y - 1$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $0$ and $0$ .

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

Step 3

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

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Step 3.1

Substitute $x^{2} + 4$ for $\frac{\partial M}{\partial y}$ and $0$ for $\frac{\partial N}{\partial x}$ .

$x^{2} + 4 = 0$

Step 3.2

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

$x^{2} + 4 = 0$ 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 $0$ for $\frac{\partial N}{\partial x}$ .

$\frac{0 - \frac{\partial M}{\partial y}}{M}$

Step 4.2

Substitute $x^{2} + 4$ for $\frac{\partial M}{\partial y}$ .

$\frac{0 - (x^{2} + 4)}{M}$

Step 4.3

Substitute $4 y + x^{2} y$ for $M$ .

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Step 4.3.1

Substitute $4 y + x^{2} y$ for $M$ .

$\frac{0 - (x^{2} + 4)}{4 y + x^{2} y}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

$\frac{0 - x^{2} - 1 \cdot 4}{4 y + x^{2} y}$

Step 4.3.2.2

Multiply $- 1$ by $4$ .

$\frac{0 - x^{2} - 4}{4 y + x^{2} y}$

Step 4.3.2.3

Subtract $- (- x^{2} - 4)$ from $0$ .

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

Step 4.3.3

Factor $y$ out of $4 y + x^{2} y$ .

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Step 4.3.3.1

Factor $y$ out of $4 y$ .

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

Step 4.3.3.2

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

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

Step 4.3.3.3

Factor $y$ out of $y \cdot 4 + y x^{2}$ .

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

Step 4.3.4

Cancel the common factor of $- x^{2} - 4$ and $4 + x^{2}$ .

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Step 4.3.4.1

Factor $- 1$ out of $- x^{2}$ .

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

Step 4.3.4.2

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- (x^{2}) - 1 (4)}{y (4 + x^{2})}$

Step 4.3.4.3

Factor $- 1$ out of $- (x^{2}) - 1 (4)$ .

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

Step 4.3.4.4

Rewrite $- (x^{2} + 4)$ as $- 1 (x^{2} + 4)$ .

$\frac{- 1 (x^{2} + 4)}{y (4 + x^{2})}$

Step 4.3.4.5

Reorder terms.

$\frac{- 1 (x^{2} + 4)}{y (x^{2} + 4)}$

Step 4.3.4.6

Cancel the common factor.

$\frac{- 1 (x^{2} + 4)}{y (x^{2} + 4)}$

Step 4.3.4.7

Rewrite the expression.

$\frac{- 1}{y}$

Step 4.3.5

Substitute $4 y + x^{2} y$ for $M$ .

$- \frac{1}{y}$

Step 4.4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

$μ (x, y) = e^{- (\ln (| y |) + C)}$

Step 5.3

Simplify.

$μ (x, y) = e^{- \ln (| y |)} + C$

Step 5.4

Simplify each term.

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Step 5.4.1

Simplify $- \ln (| y |)$ by moving $- 1$ inside the logarithm.

$μ (x, y) = e^{\ln ({| y |}^{- 1})} + C$

Step 5.4.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| y |}^{- 1} + C$

Step 5.4.3

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

$μ (x, y) = \frac{1}{| y |} + C$

Step 6

Multiply both sides of $(4 y + x^{2} y) d x + (y - 1) d y = 0$ by the integration factor $\frac{1}{y}$ .

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Step 6.1

Multiply $4 y + x^{2} y$ by $\frac{1}{y}$ .

$(4 y + x^{2} y) \frac{1}{y} d x + (y - 1) d y = 0$

Step 6.2

Multiply $4 y + x^{2} y$ by $\frac{1}{y}$ .

$\frac{4 y + x^{2} y}{y} d x + (y - 1) d y = 0$

Step 6.3

Factor $y$ out of $4 y + x^{2} y$ .

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Step 6.3.1

Factor $y$ out of $4 y$ .

$\frac{y \cdot 4 + x^{2} y}{y} d x + (y - 1) d y = 0$

Step 6.3.2

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

$\frac{y \cdot 4 + y x^{2}}{y} d x + (y - 1) d y = 0$

Step 6.3.3

Factor $y$ out of $y \cdot 4 + y x^{2}$ .

$\frac{y (4 + x^{2})}{y} d x + (y - 1) d y = 0$

Step 6.4

Cancel the common factor of $y$ .

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Step 6.4.1

Cancel the common factor.

$\frac{y (4 + x^{2})}{y} d x + (y - 1) d y = 0$

Step 6.4.2

Divide $4 + x^{2}$ by $1$ .

$(4 + x^{2}) d x + (y - 1) d y = 0$

Step 6.5

Multiply $y - 1$ by $\frac{1}{y}$ .

$(4 + x^{2}) d x + (y - 1) \frac{1}{y} d y = 0$

Step 6.6

Multiply $y - 1$ by $\frac{1}{y}$ .

$(4 + x^{2}) d x + \frac{y - 1}{y} d y = 0$

Step 7

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

$f (x, y) = \int 4 + x^{2} d x$

Step 8

Integrate $M (x, y) = 4 + x^{2}$ to find $f (x, y)$ .

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Step 8.1

Split the single integral into multiple integrals.

$f (x, y) = \int 4 d x + \int x^{2} d x$

Step 8.2

Apply the constant rule.

$f (x, y) = 4 x + C + \int x^{2} d x$

Step 8.3

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

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

Step 8.4

Simplify.

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

Step 9

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

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

Step 10

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

$\frac{\partial f}{\partial y} = \frac{y - 1}{y}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [4 x + \frac{1}{3} x^{3} + g (y)] = \frac{y - 1}{y}$

Step 11.2

By the Sum Rule, the derivative of $4 x + \frac{1}{3} x^{3} + g (y)$ with respect to $y$ is $\frac{d}{d y} [4 x] + \frac{d}{d y} [\frac{1}{3} x^{3}] + \frac{d}{d y} [g (y)]$ .

$\frac{d}{d y} [4 x] + \frac{d}{d y} [\frac{1}{3} x^{3}] + \frac{d}{d y} [g (y)] = \frac{y - 1}{y}$

Step 11.3

Since $4 x$ is constant with respect to $y$ , the derivative of $4 x$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [\frac{1}{3} x^{3}] + \frac{d}{d y} [g (y)] = \frac{y - 1}{y}$

Step 11.4

Since $\frac{1}{3} x^{3}$ is constant with respect to $y$ , the derivative of $\frac{1}{3} x^{3}$ with respect to $y$ is $0$ .

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

Step 11.5

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

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

Step 11.6

Combine terms.

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Step 11.6.1

Add $0$ and $0$ .

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

Step 11.6.2

Add $0$ and $\frac{d g}{d y}$ .

$\frac{d g}{d y} = \frac{y - 1}{y}$

Step 12

Find the antiderivative of $\frac{y - 1}{y}$ to find $g (y)$ .

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Step 12.1

Integrate both sides of $\frac{d g}{d y} = \frac{y - 1}{y}$ .

$\int \frac{d g}{d y} d y = \int \frac{y - 1}{y} d y$

Step 12.2

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

$g (y) = \int \frac{y - 1}{y} d y$

Step 12.3

Split the fraction into multiple fractions.

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

Step 12.4

Split the single integral into multiple integrals.

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

Step 12.5

Simplify.

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Step 12.5.1

Cancel the common factor of $y$ .

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Step 12.5.1.1

Cancel the common factor.

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

Step 12.5.1.2

Rewrite the expression.

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

Step 12.5.2

Move the negative in front of the fraction.

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

Step 12.6

Apply the constant rule.

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

Step 12.7

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

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

Step 12.8

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

$g (y) = y + C - (\ln (| y |) + C)$

Step 12.9

Simplify.

$g (y) = y - \ln (| y |) + C$

Step 13

Substitute for $g (y)$ in $f (x, y) = 4 x + \frac{1}{3} x^{3} + g (y)$ .

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

Step 14

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

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