Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Multiply $2$ by $1$ .

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

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

$\frac{\partial N}{\partial x} = 0 - (2 x)$

Step 2.3.3

Multiply $2$ by $- 1$ .

$\frac{\partial N}{\partial x} = 0 - 2 x$

Step 2.4

Subtract $2 x$ from $0$ .

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

Step 3

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

Tap for more steps...

Step 3.1

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

$2 x = - 2 x$

Step 3.2

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

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

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

Step 4.2

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

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

Step 4.3

Substitute $2 x y$ for $M$ .

Tap for more steps...

Step 4.3.1

Substitute $2 x y$ for $M$ .

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

Step 4.3.2

Simplify the numerator.

Tap for more steps...

Step 4.3.2.1

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

Tap for more steps...

Step 4.3.2.1.1

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

$\frac{x \cdot - 2 - 1 \cdot 2 x}{2 x y}$

Step 4.3.2.1.2

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

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

Step 4.3.2.1.3

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

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

Step 4.3.2.2

Multiply $- 1$ by $2$ .

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

Step 4.3.2.3

Subtract $2$ from $- 2$ .

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

Step 4.3.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 4.3.3.1

Cancel the common factor.

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

Step 4.3.3.2

Rewrite the expression.

$\frac{- 4}{2 y}$

Step 4.3.4

Cancel the common factor of $- 4$ and $2$ .

Tap for more steps...

Step 4.3.4.1

Factor $2$ out of $- 4$ .

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

Step 4.3.4.2

Cancel the common factors.

Tap for more steps...

Step 4.3.4.2.1

Factor $2$ out of $2 y$ .

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

Step 4.3.4.2.2

Cancel the common factor.

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

Step 4.3.4.2.3

Rewrite the expression.

$\frac{- 2}{y}$

Step 4.3.5

Substitute $2 x y$ for $M$ .

$- \frac{2}{y}$

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

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

Step 5.2

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 5.3

Multiply $2$ by $- 1$ .

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

Step 5.4

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

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

Step 5.5

Simplify.

$μ (x, y) = e^{- 2 \ln (| y |)} + C$

Step 5.6

Simplify each term.

Tap for more steps...

Step 5.6.1

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

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

Step 5.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| y |}^{- 2} + C$

Step 5.6.3

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

$μ (x, y) = y^{- 2} + C$

Step 5.6.4

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

$μ (x, y) = \frac{1}{y^{2}} + C$

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

Cancel the common factor of $y$ .

Tap for more steps...

Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Cancel the common factor.

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

Step 6.2.4

Rewrite the expression.

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Move $2$ to the left of $x$ .

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

Step 6.6

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

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

Step 6.7

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

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

Step 6.8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = x$ .

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

Step 7

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

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

Step 8

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

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Simplify the answer.

Tap for more steps...

Step 8.3.1

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

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

Step 8.3.2

Simplify.

Tap for more steps...

Step 8.3.2.1

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

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

Step 8.3.2.2

Move $2$ to the left of $y$ .

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

Step 8.3.2.3

Multiply $2$ by $y$ .

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

Step 8.3.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.3.2.4.1

Cancel the common factor.

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

Step 8.3.2.4.2

Rewrite the expression.

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

Step 8.3.2.5

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

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

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

Step 11.3

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

Tap for more steps...

Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.4

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

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

Step 11.5

Simplify.

Tap for more steps...

Step 11.5.1

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

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

Step 11.5.2

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

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

Step 11.5.3

Reorder terms.

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

Step 12

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

Tap for more steps...

Step 12.1

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

Tap for more steps...

Step 12.1.1

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

Tap for more steps...

Step 12.1.1.1

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

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

Step 12.1.1.2

Combine the numerators over the common denominator.

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

Step 12.1.1.3

Simplify each term.

Tap for more steps...

Step 12.1.1.3.1

Apply the distributive property.

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

Step 12.1.1.3.2

Expand $(- y - x) (y - x)$ using the FOIL Method.

Tap for more steps...

Step 12.1.1.3.2.1

Apply the distributive property.

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

Step 12.1.1.3.2.2

Apply the distributive property.

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

Step 12.1.1.3.2.3

Apply the distributive property.

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

Step 12.1.1.3.3

Simplify and combine like terms.

Tap for more steps...

Step 12.1.1.3.3.1

Simplify each term.

Tap for more steps...

Step 12.1.1.3.3.1.1

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

Tap for more steps...

Step 12.1.1.3.3.1.1.1

Move $y$ .

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

Step 12.1.1.3.3.1.1.2

Multiply $y$ by $y$ .

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

Step 12.1.1.3.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 12.1.1.3.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 12.1.1.3.3.1.4

Multiply $y$ by $1$ .

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

Step 12.1.1.3.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 12.1.1.3.3.1.6

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

Tap for more steps...

Step 12.1.1.3.3.1.6.1

Move $x$ .

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

Step 12.1.1.3.3.1.6.2

Multiply $x$ by $x$ .

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

Step 12.1.1.3.3.1.7

Multiply $- 1$ by $- 1$ .

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

Step 12.1.1.3.3.1.8

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

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

Step 12.1.1.3.3.2

Subtract $x y$ from $y x$ .

Tap for more steps...

Step 12.1.1.3.3.2.1

Reorder $y$ and $x$ .

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

Step 12.1.1.3.3.2.2

Subtract $x y$ from $x y$ .

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

Step 12.1.1.3.3.3

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

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

Step 12.1.1.4

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

Tap for more steps...

Step 12.1.1.4.1

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

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

Step 12.1.1.4.2

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

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

Step 12.1.1.5

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

Tap for more steps...

Step 12.1.1.5.1

Cancel the common factor.

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

Step 12.1.1.5.2

Divide $- 1$ by $1$ .

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

Step 12.1.2

Add $1$ to both sides of the equation.

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

Step 13

Find the antiderivative of $1$ to find $g (y)$ .

Tap for more steps...

Step 13.1

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

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

Step 13.2

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

$g (y) = \int d y$

Step 13.3

Apply the constant rule.

$g (y) = y + C$

Step 14

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

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