Calculus Examples

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

Step 1

Rewrite the differential equation to fit the Exact differential equation technique.

Tap for more steps...

Step 1.1

Rewrite.

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

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $M$ with respect to $y$ .

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

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

$\frac{\partial M}{\partial y} = 0 - (2 y)$

Step 2.3.3

Multiply $2$ by $- 1$ .

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

Step 2.4

Subtract $2 y$ from $0$ .

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

Step 3

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

Tap for more steps...

Step 3.1

Differentiate $N$ with respect to $x$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $1$ .

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

Step 4

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

Tap for more steps...

Step 4.1

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

$- 2 y = 2 y$

Step 4.2

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

$- 2 y = 2 y$ is not an identity.

Step 5

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

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Substitute $2 x y$ for $N$ .

Tap for more steps...

Step 5.3.1

Substitute $2 x y$ for $N$ .

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

Step 5.3.2

Simplify the numerator.

Tap for more steps...

Step 5.3.2.1

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

Tap for more steps...

Step 5.3.2.1.1

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

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

Step 5.3.2.1.2

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

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

Step 5.3.2.1.3

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

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

Step 5.3.2.2

Multiply $- 1$ by $2$ .

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

Step 5.3.2.3

Subtract $2$ from $- 2$ .

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

Step 5.3.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 5.3.3.1

Cancel the common factor.

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

Step 5.3.3.2

Rewrite the expression.

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

Step 5.3.4

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

Tap for more steps...

Step 5.3.4.1

Factor $2$ out of $- 4$ .

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

Step 5.3.4.2

Cancel the common factors.

Tap for more steps...

Step 5.3.4.2.1

Factor $2$ out of $2 x$ .

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

Step 5.3.4.2.2

Cancel the common factor.

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

Step 5.3.4.2.3

Rewrite the expression.

$\frac{- 2}{x}$

Step 5.3.5

Move the negative in front of the fraction.

$- \frac{2}{x}$

Step 5.4

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 6

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

Tap for more steps...

Step 6.1

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 6.2

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 6.3

Multiply $2$ by $- 1$ .

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

Step 6.4

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

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

Step 6.5

Simplify.

$μ (x, y) = e^{- 2 \ln (| x |)} + C$

Step 6.6

Simplify each term.

Tap for more steps...

Step 6.6.1

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

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

Step 6.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| x |}^{- 2} + C$

Step 6.6.3

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

$μ (x, y) = x^{- 2} + C$

Step 6.6.4

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

$μ (x, y) = \frac{1}{x^{2}} + C$

Step 7

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

Tap for more steps...

Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Cancel the common factor of $x$ .

Tap for more steps...

Step 7.5.1

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

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

Step 7.5.2

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

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

Step 7.5.3

Cancel the common factor.

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

Step 7.5.4

Rewrite the expression.

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

Step 7.6

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

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

Step 7.7

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

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

Step 7.8

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

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

Step 8

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

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

Step 9

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

Tap for more steps...

Step 9.1

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

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

Step 9.2

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

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

Step 9.3

Simplify the answer.

Tap for more steps...

Step 9.3.1

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

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

Step 9.3.2

Simplify.

Tap for more steps...

Step 9.3.2.1

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

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

Step 9.3.2.2

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

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

Step 9.3.2.3

Multiply $2$ by $x$ .

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

Step 9.3.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.3.2.4.1

Cancel the common factor.

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

Step 9.3.2.4.2

Rewrite the expression.

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

Step 9.3.2.5

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

Tap for more steps...

Step 12.1

Differentiate $f$ with respect to $x$ .

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

Step 12.2

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

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

Step 12.3

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

Tap for more steps...

Step 12.3.1

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

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

Step 12.3.2

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

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

Step 12.3.3

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

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

Step 12.4

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

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

Step 12.5

Simplify.

Tap for more steps...

Step 12.5.1

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

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

Step 12.5.2

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

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

Step 12.5.3

Reorder terms.

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

Step 13

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

Tap for more steps...

Step 13.1

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

Tap for more steps...

Step 13.1.1

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

Tap for more steps...

Step 13.1.1.1

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

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

Step 13.1.1.2

Combine the numerators over the common denominator.

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

Step 13.1.1.3

Simplify each term.

Tap for more steps...

Step 13.1.1.3.1

Apply the distributive property.

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

Step 13.1.1.3.2

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

Tap for more steps...

Step 13.1.1.3.2.1

Apply the distributive property.

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

Step 13.1.1.3.2.2

Apply the distributive property.

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

Step 13.1.1.3.2.3

Apply the distributive property.

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

Step 13.1.1.3.3

Simplify and combine like terms.

Tap for more steps...

Step 13.1.1.3.3.1

Simplify each term.

Tap for more steps...

Step 13.1.1.3.3.1.1

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

Tap for more steps...

Step 13.1.1.3.3.1.1.1

Move $x$ .

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

Step 13.1.1.3.3.1.1.2

Multiply $x$ by $x$ .

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

Step 13.1.1.3.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 13.1.1.3.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 13.1.1.3.3.1.4

Multiply $x$ by $1$ .

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

Step 13.1.1.3.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 13.1.1.3.3.1.6

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

Tap for more steps...

Step 13.1.1.3.3.1.6.1

Move $y$ .

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

Step 13.1.1.3.3.1.6.2

Multiply $y$ by $y$ .

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

Step 13.1.1.3.3.1.7

Multiply $- 1$ by $- 1$ .

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

Step 13.1.1.3.3.1.8

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

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

Step 13.1.1.3.3.2

Subtract $y x$ from $x y$ .

Tap for more steps...

Step 13.1.1.3.3.2.1

Move $y$ .

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

Step 13.1.1.3.3.2.2

Subtract $x y$ from $x y$ .

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

Step 13.1.1.3.3.3

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

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

Step 13.1.1.4

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

Tap for more steps...

Step 13.1.1.4.1

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

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

Step 13.1.1.4.2

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

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

Step 13.1.1.5

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

Tap for more steps...

Step 13.1.1.5.1

Cancel the common factor.

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

Step 13.1.1.5.2

Divide $- 1$ by $1$ .

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

Step 13.1.2

Add $1$ to both sides of the equation.

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

Step 14

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

Tap for more steps...

Step 14.1

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

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

Step 14.2

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

$g (x) = \int d x$

Step 14.3

Apply the constant rule.

$g (x) = x + C$

Step 15

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

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