Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

$\frac{\partial M}{\partial y} = \frac{d}{d y} [y^{- 1}]$

Step 1.3

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

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

Step 1.4

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

$\frac{\partial M}{\partial y} = - \frac{1}{y^{2}}$

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- (3 y - \frac{x}{y^{2}})$ with respect to $x$ is $- \frac{d}{d x} [3 y - \frac{x}{y^{2}}]$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $0$ and $\frac{d}{d x} [- \frac{x}{y^{2}}]$ .

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

Step 2.6

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

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

Step 2.7

Multiply.

Tap for more steps...

Step 2.7.1

Multiply $- 1$ by $- 1$ .

$\frac{\partial N}{\partial x} = 1 (\frac{1}{y^{2}} \frac{d}{d x} [x])$

Step 2.7.2

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 3

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

Tap for more steps...

Step 3.1

Substitute $- \frac{1}{y^{2}}$ for $\frac{\partial M}{\partial y}$ and $\frac{1}{y^{2}}$ for $\frac{\partial N}{\partial x}$ .

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

Step 3.2

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

$- \frac{1}{y^{2}} = \frac{1}{y^{2}}$ is not an identity.

Step 4

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

Tap for more steps...

Step 4.1

Substitute $\frac{1}{y^{2}}$ for $\frac{\partial N}{\partial x}$ .

$\frac{\frac{1}{y^{2}} - \frac{\partial M}{\partial y}}{M}$

Step 4.2

Substitute $- \frac{1}{y^{2}}$ for $\frac{\partial M}{\partial y}$ .

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

Step 4.3

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

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.3

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

Tap for more steps...

Step 4.3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3.2

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

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

Step 4.3.4

Combine the numerators over the common denominator.

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

Step 4.3.5

Add $1$ and $1$ .

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

Step 4.3.6

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

Tap for more steps...

Step 4.3.6.1

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

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

Step 4.3.6.2

Cancel the common factor.

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

Step 4.3.6.3

Rewrite the expression.

$\frac{2}{y}$

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

Tap for more steps...

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.

Tap for more steps...

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 $\frac{1}{y} d x - (3 y - \frac{x}{y^{2}}) d y = 0$ by the integration factor $y^{2}$ .

Tap for more steps...

Step 6.1

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

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

Step 6.2

Cancel the common factor of $y$ .

Tap for more steps...

Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 6.3

Multiply $- (3 y - \frac{x}{y^{2}})$ by $y^{2}$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $3$ by $- 1$ .

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

Step 6.6

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

Tap for more steps...

Step 6.6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.6.2

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

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

Step 6.7

Apply the distributive property.

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

Step 6.8

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

Tap for more steps...

Step 6.8.1

Move $y^{2}$ .

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

Step 6.8.2

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

Tap for more steps...

Step 6.8.2.1

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

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

Step 6.8.2.2

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

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

Step 6.8.3

Add $2$ and $1$ .

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

Step 6.9

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

Tap for more steps...

Step 6.9.1

Cancel the common factor.

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

Step 6.9.2

Rewrite the expression.

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

Step 7

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

$f (x, y) = \int y d x$

Step 8

Integrate $M (x, y) = y$ to find $f (x, y)$ .

Tap for more steps...

Step 8.1

Apply the constant rule.

$f (x, y) = y x + C$

Step 9

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

$f (x, y) = y x + g (y)$

Step 10

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

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

Step 11

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

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

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

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

Step 11.3

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

Tap for more steps...

Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

Multiply $x$ by $1$ .

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

Step 12

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

Tap for more steps...

Step 12.1

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

Tap for more steps...

Step 12.1.1

Subtract $x$ from both sides of the equation.

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

Step 12.1.2

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

Tap for more steps...

Step 12.1.2.1

Subtract $x$ from $x$ .

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

Step 12.1.2.2

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

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

Step 13

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

Tap for more steps...

Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

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

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

Step 13.5

Simplify the answer.

Tap for more steps...

Step 13.5.1

Rewrite $- 3 (\frac{1}{4} y^{4} + C)$ as $- 3 (\frac{1}{4}) y^{4} + C$ .

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

Step 13.5.2

Simplify.

Tap for more steps...

Step 13.5.2.1

Combine $- 3$ and $\frac{1}{4}$ .

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

Step 13.5.2.2

Move the negative in front of the fraction.

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

Step 14

Substitute for $g (y)$ in $f (x, y) = y x + g (y)$ .

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

Step 15

Simplify each term.

Tap for more steps...

Step 15.1

Combine $y^{4}$ and $\frac{3}{4}$ .

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

Step 15.2

Move $3$ to the left of $y^{4}$ .

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