Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 3 x \cdot 1 + \frac{d}{d y} [- y^{2}]$

Step 1.3.3

Multiply $3$ by $1$ .

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

Step 1.4

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

Tap for more steps...

Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Multiply $2$ by $- 1$ .

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

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x - y$ .

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

Step 2.3

Differentiate.

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Simplify the expression.

Tap for more steps...

Step 2.3.4.1

Add $1$ and $0$ .

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

Step 2.3.4.2

Multiply $x$ by $1$ .

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

Step 2.3.5

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

$\frac{\partial N}{\partial x} = x + (x - y) \cdot 1$

Step 2.3.6

Simplify by adding terms.

Tap for more steps...

Step 2.3.6.1

Multiply $x - y$ by $1$ .

$\frac{\partial N}{\partial x} = x + x - y$

Step 2.3.6.2

Add $x$ and $x$ .

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

Step 3

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

Tap for more steps...

Step 3.1

Substitute $3 x - 2 y$ for $\frac{\partial M}{\partial y}$ and $2 x - y$ for $\frac{\partial N}{\partial x}$ .

$3 x - 2 y = 2 x - y$

Step 3.2

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

$3 x - 2 y = 2 x - y$ is not an identity.

Step 4

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 4.1

Substitute $3 x - 2 y$ for $\frac{\partial M}{\partial y}$ .

$\frac{3 x - 2 y - \frac{\partial N}{\partial x}}{N}$

Step 4.2

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

$\frac{3 x - 2 y - (2 x - y)}{N}$

Step 4.3

Substitute $x (x - y)$ for $N$ .

Tap for more steps...

Step 4.3.1

Substitute $x (x - y)$ for $N$ .

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

Step 4.3.2

Simplify the numerator.

Tap for more steps...

Step 4.3.2.1

Apply the distributive property.

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

Step 4.3.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.2.3

Multiply $- - y$ .

Tap for more steps...

Step 4.3.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2.3.2

Multiply $y$ by $1$ .

$\frac{3 x - 2 y - 2 x + y}{x (x - y)}$

Step 4.3.2.4

Subtract $2 x$ from $3 x$ .

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

Step 4.3.2.5

Add $- 2 y$ and $y$ .

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

Step 4.3.3

Cancel the common factor of $x - y$ .

Tap for more steps...

Step 4.3.3.1

Cancel the common factor.

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

Step 4.3.3.2

Rewrite the expression.

$\frac{1}{x}$

Step 4.4

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

$μ (x, y) = e^{\int \frac{1}{x} d x}$

Step 5

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

Tap for more steps...

Step 5.1

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

$μ (x, y) = e^{\ln (| x |) + C}$

Step 5.2

Simplify the answer.

Tap for more steps...

Step 5.2.1

Simplify.

$μ (x, y) = e^{\ln (| x |)} + C$

Step 5.2.2

Exponentiation and log are inverse functions.

$μ (x, y) = | x | + C$

Step 6

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

Tap for more steps...

Step 6.1

Multiply $3 x y - y^{2}$ by $x$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

Tap for more steps...

Step 6.3.1

Move $x$ .

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

Step 6.3.2

Multiply $x$ by $x$ .

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

Step 6.4

Multiply $x (x - y)$ by $x$ .

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

Step 6.5

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

Tap for more steps...

Step 6.5.1

Move $x$ .

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

Step 6.5.2

Multiply $x$ by $x$ .

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

Step 6.6

Apply the distributive property.

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

Step 6.7

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

Tap for more steps...

Step 6.7.1

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

Tap for more steps...

Step 6.7.1.1

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

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

Step 6.7.1.2

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

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

Step 6.7.2

Add $2$ and $1$ .

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

Step 6.8

Rewrite using the commutative property of multiplication.

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

Step 7

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

$f (x, y) = \int 3 x^{2} y - y^{2} x d x$

Step 8

Integrate $M (x, y) = 3 x^{2} y - y^{2} x$ to find $f (x, y)$ .

Tap for more steps...

Step 8.1

Split the single integral into multiple integrals.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

Simplify.

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

Step 8.7

Simplify.

Tap for more steps...

Step 8.7.1

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

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

Step 8.7.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.7.2.1

Cancel the common factor.

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

Step 8.7.2.2

Rewrite the expression.

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

Step 8.7.3

Multiply $y$ by $1$ .

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

Step 8.7.4

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

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

Step 8.7.5

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

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

Step 8.8

Simplify.

Tap for more steps...

Step 8.8.1

Reorder terms.

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

Step 8.8.2

Remove parentheses.

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

Step 8.8.3

Remove parentheses.

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

Step 9

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

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

Step 10

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

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

Step 11

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

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

By the Sum Rule, the derivative of $y x^{3} - \frac{1}{2} x^{2} y^{2} + g (y)$ with respect to $y$ is $\frac{d}{d y} [y x^{3}] + \frac{d}{d y} [- \frac{1}{2} x^{2} y^{2}] + \frac{d}{d y} [g (y)]$ .

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

Step 11.3

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

Tap for more steps...

Step 11.3.1

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

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

Step 11.3.2

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

$x^{3} \cdot 1 + \frac{d}{d y} [- \frac{1}{2} x^{2} y^{2}] + \frac{d}{d y} [g (y)] = x^{3} - x^{2} y$

Step 11.3.3

Multiply $x^{3}$ by $1$ .

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

Step 11.4

Evaluate $\frac{d}{d y} [- \frac{1}{2} x^{2} y^{2}]$ .

Tap for more steps...

Step 11.4.1

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

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

Step 11.4.2

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

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

Step 11.4.3

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

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

Step 11.4.4

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

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

Step 11.4.5

Multiply $2$ by $- 1$ .

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

Step 11.4.6

Combine $- 2$ and $\frac{x^{2}}{2}$ .

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

Step 11.4.7

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

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

Step 11.4.8

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

Tap for more steps...

Step 11.4.8.1

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

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

Step 11.4.8.2

Cancel the common factors.

Tap for more steps...

Step 11.4.8.2.1

Factor $2$ out of $2$ .

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

Step 11.4.8.2.2

Cancel the common factor.

$x^{3} + \frac{2 (- x^{2} y)}{2 \cdot 1} + \frac{d}{d y} [g (y)] = x^{3} - x^{2} y$

Step 11.4.8.2.3

Rewrite the expression.

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

Step 11.4.8.2.4

Divide $- x^{2} y$ by $1$ .

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

Step 11.5

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

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

Step 11.6

Reorder terms.

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

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

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

Step 12.1.2

Add $x^{2} y$ to both sides of the equation.

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

Step 12.1.3

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

Tap for more steps...

Step 12.1.3.1

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

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

Step 12.1.3.2

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

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

Step 12.1.3.3

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

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

Step 13

Find the antiderivative of $0$ to find $g (y)$ .

Tap for more steps...

Step 13.1

Integrate both sides of $\frac{d g}{d y} = 0$ .

$\int \frac{d g}{d y} d y = \int 0 d y$

Step 13.2

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

$g (y) = \int 0 d y$

Step 13.3

The integral of $0$ with respect to $y$ is $0$ .

$g (y) = 0 + C$

Step 13.4

Add $0$ and $C$ .

$g (y) = C$

Step 14

Substitute for $g (y)$ in $f (x, y) = y x^{3} - \frac{1}{2} x^{2} y^{2} + g (y)$ .

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

Step 15

Simplify each term.

Tap for more steps...

Step 15.1

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

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

Step 15.2

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

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