Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = 3 x y - 1$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $3$ by $1$ .

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

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

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

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

Step 2.4.2

Add $3 y$ and $0$ .

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

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 $3 y$ for $\frac{\partial N}{\partial x}$ .

$2 y = 3 y$

Step 3.2

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

$2 y = 3 y$ 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 $3 y$ for $\frac{\partial N}{\partial x}$ .

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

Step 4.2

Substitute $2 y$ for $\frac{\partial M}{\partial y}$ .

$\frac{3 y - (2 y)}{M}$

Step 4.3

Substitute $y^{2}$ for $M$ .

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Step 4.3.1

Substitute $y^{2}$ for $M$ .

$\frac{3 y - (2 y)}{y^{2}}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $y$ out of $3 y - 1 \cdot 2 y$ .

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Step 4.3.2.1.1

Factor $y$ out of $3 y$ .

$\frac{y \cdot 3 - 1 \cdot 2 y}{y^{2}}$

Step 4.3.2.1.2

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

$\frac{y \cdot 3 + y (- 1 \cdot 2)}{y^{2}}$

Step 4.3.2.1.3

Factor $y$ out of $y \cdot 3 + y (- 1 \cdot 2)$ .

$\frac{y (3 - 1 \cdot 2)}{y^{2}}$

Step 4.3.2.2

Multiply $- 1$ by $2$ .

$\frac{y (3 - 2)}{y^{2}}$

Step 4.3.2.3

Subtract $2$ from $3$ .

$\frac{y \cdot 1}{y^{2}}$

Step 4.3.3

Multiply $y$ by $1$ .

$\frac{y}{y^{2}}$

Step 4.3.4

Substitute $y^{2}$ for $M$ .

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Step 4.3.4.1

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

$\frac{y^{1}}{y^{2}}$

Step 4.3.4.2

Factor $y$ out of $y^{1}$ .

$\frac{y \cdot 1}{y^{2}}$

Step 4.3.4.3

Cancel the common factors.

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Step 4.3.4.3.1

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

$\frac{y \cdot 1}{y \cdot y}$

Step 4.3.4.3.2

Cancel the common factor.

$\frac{y \cdot 1}{y \cdot y}$

Step 4.3.4.3.3

Rewrite the expression.

$\frac{1}{y}$

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

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

$μ (x, y) = e^{\ln (| y |) + C}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Simplify.

$μ (x, y) = e^{\ln (| y |)} + C$

Step 5.2.2

Exponentiation and log are inverse functions.

$μ (x, y) = | y | + C$

Step 6

Multiply both sides of $y^{2} d x + (3 x y - 1) d y = 0$ by the integration factor $y$ .

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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

Step 6.2.2

Add $2$ and $1$ .

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

Step 6.3

Multiply $3 x y - 1$ by $y$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $y$ by $y$ by adding the exponents.

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Step 6.5.1

Move $y$ .

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

Step 6.5.2

Multiply $y$ by $y$ .

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

Step 6.6

Rewrite $- 1 y$ as $- y$ .

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

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

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

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

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

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

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

Step 12.1.2

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

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Step 12.1.2.1

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

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

Step 12.1.2.2

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

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

Step 12.1.2.3

Add $- y$ and $0$ .

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

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

$g (y) = - (\frac{1}{2} y^{2} + C)$

Step 13.5

Rewrite $- (\frac{1}{2} y^{2} + C)$ as $- \frac{1}{2} y^{2} + C$ .

$g (y) = - \frac{1}{2} y^{2} + C$

Step 14

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

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

Step 15

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

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