Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Since $- 2$ is constant with respect to $y$ , the derivative of $- 2$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 1 + 0$

Step 1.5

Add $1$ and $0$ .

$\frac{\partial M}{\partial y} = 1$

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $3$ by $1$ .

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

Step 2.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

Add $3$ and $0$ .

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

Step 3

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

Tap for more steps...

Step 3.1

Substitute $1$ for $\frac{\partial M}{\partial y}$ and $3$ for $\frac{\partial N}{\partial x}$ .

$1 = 3$

Step 3.2

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

$1 = 3$ 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 $3$ for $\frac{\partial N}{\partial x}$ .

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

Step 4.2

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

$\frac{3 - 1}{M}$

Step 4.3

Substitute $y - 2$ for $M$ .

Tap for more steps...

Step 4.3.1

Substitute $y - 2$ for $M$ .

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

Step 4.3.2

Substitute $y - 2$ for $M$ .

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

Step 4.4

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

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

Step 5

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

Tap for more steps...

Step 5.1

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

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

Step 5.2

Let $u = y - 2$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 5.2.1

Let $u = y - 2$ . Find $\frac{d u}{d y}$ .

Tap for more steps...

Step 5.2.1.1

Differentiate $y - 2$ .

$\frac{d}{d y} [y - 2]$

Step 5.2.1.2

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

$\frac{d}{d y} [y] + \frac{d}{d y} [- 2]$

Step 5.2.1.3

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

$1 + \frac{d}{d y} [- 2]$

Step 5.2.1.4

Since $- 2$ is constant with respect to $y$ , the derivative of $- 2$ with respect to $y$ is $0$ .

$1 + 0$

Step 5.2.1.5

Add $1$ and $0$ .

$1$

Step 5.2.2

Rewrite the problem using $u$ and $d u$ .

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

Step 5.3

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

$μ (x, y) = e^{2 (\ln (| u |) + C)}$

Step 5.4

Simplify.

$μ (x, y) = e^{2 \ln (| u |)} + C$

Step 5.5

Replace all occurrences of $u$ with $y - 2$ .

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

Step 5.6

Simplify each term.

Tap for more steps...

Step 5.6.1

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

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

Step 5.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| y - 2 |}^{2} + C$

Step 5.6.3

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

$μ (x, y) = {(y - 2)}^{2} + C$

Step 5.6.4

Rewrite ${(y - 2)}^{2}$ as $(y - 2) (y - 2)$ .

$μ (x, y) = (y - 2) (y - 2) + C$

Step 5.6.5

Expand $(y - 2) (y - 2)$ using the FOIL Method.

Tap for more steps...

Step 5.6.5.1

Apply the distributive property.

$μ (x, y) = y (y - 2) - 2 (y - 2) + C$

Step 5.6.5.2

Apply the distributive property.

$μ (x, y) = y \cdot y + y \cdot - 2 - 2 (y - 2) + C$

Step 5.6.5.3

Apply the distributive property.

$μ (x, y) = y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2 + C$

Step 5.6.6

Simplify and combine like terms.

Tap for more steps...

Step 5.6.6.1

Simplify each term.

Tap for more steps...

Step 5.6.6.1.1

Multiply $y$ by $y$ .

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

Step 5.6.6.1.2

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

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

Step 5.6.6.1.3

Multiply $- 2$ by $- 2$ .

$μ (x, y) = y^{2} - 2 y - 2 y + 4 + C$

Step 5.6.6.2

Subtract $2 y$ from $- 2 y$ .

$μ (x, y) = y^{2} - 4 y + 4 + C$

Step 6

Multiply both sides of $(y - 2) d x + (3 x - y) d y = 0$ by the integration factor $y^{2} - 4 y + 4$ .

Tap for more steps...

Step 6.1

Multiply $y - 2$ by $y^{2} - 4 y + 4$ .

$(y - 2) (y^{2} - 4 y + 4) d x + (3 x - y) d y = 0$

Step 6.2

Expand $(y - 2) (y^{2} - 4 y + 4)$ by multiplying each term in the first expression by each term in the second expression.

$(y \cdot y^{2} + y (- 4 y) + y \cdot 4 - 2 y^{2} - 2 (- 4 y) - 2 \cdot 4) d x + (3 x - y) d y = 0$

Step 6.3

Simplify each term.

Tap for more steps...

Step 6.3.1

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

Tap for more steps...

Step 6.3.1.1

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

Tap for more steps...

Step 6.3.1.1.1

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

$(y^{1} y^{2} + y (- 4 y) + y \cdot 4 - 2 y^{2} - 2 (- 4 y) - 2 \cdot 4) d x + (3 x - y) d y = 0$

Step 6.3.1.1.2

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

$(y^{1 + 2} + y (- 4 y) + y \cdot 4 - 2 y^{2} - 2 (- 4 y) - 2 \cdot 4) d x + (3 x - y) d y = 0$

Step 6.3.1.2

Add $1$ and $2$ .

$(y^{3} + y (- 4 y) + y \cdot 4 - 2 y^{2} - 2 (- 4 y) - 2 \cdot 4) d x + (3 x - y) d y = 0$

Step 6.3.2

Rewrite using the commutative property of multiplication.

$(y^{3} - 4 y \cdot y + y \cdot 4 - 2 y^{2} - 2 (- 4 y) - 2 \cdot 4) d x + (3 x - y) d y = 0$

Step 6.3.3

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

Tap for more steps...

Step 6.3.3.1

Move $y$ .

$(y^{3} - 4 (y \cdot y) + y \cdot 4 - 2 y^{2} - 2 (- 4 y) - 2 \cdot 4) d x + (3 x - y) d y = 0$

Step 6.3.3.2

Multiply $y$ by $y$ .

$(y^{3} - 4 y^{2} + y \cdot 4 - 2 y^{2} - 2 (- 4 y) - 2 \cdot 4) d x + (3 x - y) d y = 0$

Step 6.3.4

Move $4$ to the left of $y$ .

$(y^{3} - 4 y^{2} + 4 \cdot y - 2 y^{2} - 2 (- 4 y) - 2 \cdot 4) d x + (3 x - y) d y = 0$

Step 6.3.5

Multiply $- 4$ by $- 2$ .

$(y^{3} - 4 y^{2} + 4 y - 2 y^{2} + 8 y - 2 \cdot 4) d x + (3 x - y) d y = 0$

Step 6.3.6

Multiply $- 2$ by $4$ .

$(y^{3} - 4 y^{2} + 4 y - 2 y^{2} + 8 y - 8) d x + (3 x - y) d y = 0$

Step 6.4

Subtract $2 y^{2}$ from $- 4 y^{2}$ .

$(y^{3} - 6 y^{2} + 4 y + 8 y - 8) d x + (3 x - y) d y = 0$

Step 6.5

Add $4 y$ and $8 y$ .

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x - y) d y = 0$

Step 6.6

Multiply $3 x - y$ by $y^{2} - 4 y + 4$ .

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x - y) (y^{2} - 4 y + 4) d y = 0$

Step 6.7

Expand $(3 x - y) (y^{2} - 4 y + 4)$ by multiplying each term in the first expression by each term in the second expression.

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} + 3 x (- 4 y) + 3 x \cdot 4 - y \cdot y^{2} - y (- 4 y) - y \cdot 4) d y = 0$

Step 6.8

Simplify each term.

Tap for more steps...

Step 6.8.1

Rewrite using the commutative property of multiplication.

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} + 3 \cdot - 4 x y + 3 x \cdot 4 - y \cdot y^{2} - y (- 4 y) - y \cdot 4) d y = 0$

Step 6.8.2

Multiply $3$ by $- 4$ .

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 3 x \cdot 4 - y \cdot y^{2} - y (- 4 y) - y \cdot 4) d y = 0$

Step 6.8.3

Multiply $4$ by $3$ .

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 12 x - y \cdot y^{2} - y (- 4 y) - y \cdot 4) d y = 0$

Step 6.8.4

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

Tap for more steps...

Step 6.8.4.1

Move $y^{2}$ .

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 12 x - (y^{2} y) - y (- 4 y) - y \cdot 4) d y = 0$

Step 6.8.4.2

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

Tap for more steps...

Step 6.8.4.2.1

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

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 12 x - (y^{2} y^{1}) - y (- 4 y) - y \cdot 4) d y = 0$

Step 6.8.4.2.2

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

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 12 x - y^{2 + 1} - y (- 4 y) - y \cdot 4) d y = 0$

Step 6.8.4.3

Add $2$ and $1$ .

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 12 x - y^{3} - y (- 4 y) - y \cdot 4) d y = 0$

Step 6.8.5

Rewrite using the commutative property of multiplication.

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 12 x - y^{3} - 1 \cdot - 4 y \cdot y - y \cdot 4) d y = 0$

Step 6.8.6

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

Tap for more steps...

Step 6.8.6.1

Move $y$ .

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 12 x - y^{3} - 1 \cdot - 4 (y \cdot y) - y \cdot 4) d y = 0$

Step 6.8.6.2

Multiply $y$ by $y$ .

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 12 x - y^{3} - 1 \cdot - 4 y^{2} - y \cdot 4) d y = 0$

Step 6.8.7

Multiply $- 1$ by $- 4$ .

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - y \cdot 4) d y = 0$

Step 6.8.8

Multiply $4$ by $- 1$ .

$(y^{3} - 6 y^{2} + 12 y - 8) d x + (3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y) d y = 0$

Step 7

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

$f (x, y) = \int y^{3} - 6 y^{2} + 12 y - 8 d x$

Step 8

Integrate $M (x, y) = y^{3} - 6 y^{2} + 12 y - 8$ to find $f (x, y)$ .

Tap for more steps...

Step 8.1

Apply the constant rule.

$f (x, y) = (y^{3} - 6 y^{2} + 12 y - 8) x + C$

Step 9

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

$f (x, y) = (y^{3} - 6 y^{2} + 12 y - 8) x + g (y)$

Step 10

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

$\frac{\partial f}{\partial y} = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11

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

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [(y^{3} - 6 y^{2} + 12 y - 8) x + g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.2

By the Sum Rule, the derivative of $(y^{3} - 6 y^{2} + 12 y - 8) x + g (y)$ with respect to $y$ is $\frac{d}{d y} [(y^{3} - 6 y^{2} + 12 y - 8) x] + \frac{d}{d y} [g (y)]$ .

$\frac{d}{d y} [(y^{3} - 6 y^{2} + 12 y - 8) x] + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3

Evaluate $\frac{d}{d y} [(y^{3} - 6 y^{2} + 12 y - 8) x]$ .

Tap for more steps...

Step 11.3.1

Since $x$ is constant with respect to $y$ , the derivative of $(y^{3} - 6 y^{2} + 12 y - 8) x$ with respect to $y$ is $x \frac{d}{d y} [y^{3} - 6 y^{2} + 12 y - 8]$ .

$x \frac{d}{d y} [y^{3} - 6 y^{2} + 12 y - 8] + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3.2

By the Sum Rule, the derivative of $y^{3} - 6 y^{2} + 12 y - 8$ with respect to $y$ is $\frac{d}{d y} [y^{3}] + \frac{d}{d y} [- 6 y^{2}] + \frac{d}{d y} [12 y] + \frac{d}{d y} [- 8]$ .

$x (\frac{d}{d y} [y^{3}] + \frac{d}{d y} [- 6 y^{2}] + \frac{d}{d y} [12 y] + \frac{d}{d y} [- 8]) + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3.3

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

$x (3 y^{2} + \frac{d}{d y} [- 6 y^{2}] + \frac{d}{d y} [12 y] + \frac{d}{d y} [- 8]) + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3.4

Since $- 6$ is constant with respect to $y$ , the derivative of $- 6 y^{2}$ with respect to $y$ is $- 6 \frac{d}{d y} [y^{2}]$ .

$x (3 y^{2} - 6 \frac{d}{d y} [y^{2}] + \frac{d}{d y} [12 y] + \frac{d}{d y} [- 8]) + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3.5

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

$x (3 y^{2} - 6 (2 y) + \frac{d}{d y} [12 y] + \frac{d}{d y} [- 8]) + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3.6

Since $12$ is constant with respect to $y$ , the derivative of $12 y$ with respect to $y$ is $12 \frac{d}{d y} [y]$ .

$x (3 y^{2} - 6 (2 y) + 12 \frac{d}{d y} [y] + \frac{d}{d y} [- 8]) + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3.7

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

$x (3 y^{2} - 6 (2 y) + 12 \cdot 1 + \frac{d}{d y} [- 8]) + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3.8

Since $- 8$ is constant with respect to $y$ , the derivative of $- 8$ with respect to $y$ is $0$ .

$x (3 y^{2} - 6 (2 y) + 12 \cdot 1 + 0) + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3.9

Multiply $2$ by $- 6$ .

$x (3 y^{2} - 12 y + 12 \cdot 1 + 0) + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3.10

Multiply $12$ by $1$ .

$x (3 y^{2} - 12 y + 12 + 0) + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.3.11

Add $3 y^{2} - 12 y + 12$ and $0$ .

$x (3 y^{2} - 12 y + 12) + \frac{d}{d y} [g (y)] = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.4

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

$x (3 y^{2} - 12 y + 12) + \frac{d g}{d y} = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.5

Simplify.

Tap for more steps...

Step 11.5.1

Apply the distributive property.

$x (3 y^{2}) + x (- 12 y) + x \cdot 12 + \frac{d g}{d y} = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.5.2

Combine terms.

Tap for more steps...

Step 11.5.2.1

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

$3 \cdot x y^{2} + x (- 12 y) + x \cdot 12 + \frac{d g}{d y} = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.5.2.2

Move $12$ to the left of $x$ .

$3 x y^{2} + x (- 12 y) + 12 x + \frac{d g}{d y} = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

Step 11.5.3

Reorder terms.

$\frac{d g}{d y} + 3 y^{2} x - 12 x y + 12 x = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$

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

$\frac{d g}{d y} - 12 x y + 12 x = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y - 3 y^{2} x$

Step 12.1.2

Add $12 x y$ to both sides of the equation.

$\frac{d g}{d y} + 12 x = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y - 3 y^{2} x + 12 x y$

Step 12.1.3

Subtract $12 x$ from both sides of the equation.

$\frac{d g}{d y} = 3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y - 3 y^{2} x + 12 x y - 12 x$

Step 12.1.4

Combine the opposite terms in $3 x y^{2} - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y - 3 y^{2} x + 12 x y - 12 x$ .

Tap for more steps...

Step 12.1.4.1

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

$\frac{d g}{d y} = 3 y^{2} x - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y - 3 y^{2} x + 12 x y - 12 x$

Step 12.1.4.2

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

$\frac{d g}{d y} = - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y + 0 + 12 x y - 12 x$

Step 12.1.4.3

Add $- 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y$ and $0$ .

$\frac{d g}{d y} = - 12 x y + 12 x - y^{3} + 4 y^{2} - 4 y + 12 x y - 12 x$

Step 12.1.4.4

Add $- 12 x y$ and $12 x y$ .

$\frac{d g}{d y} = 12 x - y^{3} + 4 y^{2} - 4 y + 0 - 12 x$

Step 12.1.4.5

Add $12 x - y^{3} + 4 y^{2} - 4 y$ and $0$ .

$\frac{d g}{d y} = 12 x - y^{3} + 4 y^{2} - 4 y - 12 x$

Step 12.1.4.6

Subtract $12 x$ from $12 x$ .

$\frac{d g}{d y} = - y^{3} + 4 y^{2} - 4 y + 0$

Step 12.1.4.7

Add $- y^{3} + 4 y^{2} - 4 y$ and $0$ .

$\frac{d g}{d y} = - y^{3} + 4 y^{2} - 4 y$

Step 13

Find the antiderivative of $- y^{3} + 4 y^{2} - 4 y$ to find $g (y)$ .

Tap for more steps...

Step 13.1

Integrate both sides of $\frac{d g}{d y} = - y^{3} + 4 y^{2} - 4 y$ .

$\int \frac{d g}{d y} d y = \int - y^{3} + 4 y^{2} - 4 y d y$

Step 13.2

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

$g (y) = \int - y^{3} + 4 y^{2} - 4 y d y$

Step 13.3

Split the single integral into multiple integrals.

$g (y) = \int - y^{3} d y + \int 4 y^{2} d y + \int - 4 y d y$

Step 13.4

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

$g (y) = - \int y^{3} d y + \int 4 y^{2} d y + \int - 4 y d y$

Step 13.5

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

$g (y) = - (\frac{1}{4} y^{4} + C) + \int 4 y^{2} d y + \int - 4 y d y$

Step 13.6

Since $4$ is constant with respect to $y$ , move $4$ out of the integral.

$g (y) = - (\frac{1}{4} y^{4} + C) + 4 \int y^{2} d y + \int - 4 y d y$

Step 13.7

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

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

Step 13.8

Since $- 4$ is constant with respect to $y$ , move $- 4$ out of the integral.

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

Step 13.9

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

$g (y) = - (\frac{1}{4} y^{4} + C) + 4 (\frac{1}{3} y^{3} + C) - 4 (\frac{1}{2} y^{2} + C)$

Step 13.10

Simplify.

$g (y) = - \frac{y^{4}}{4} + \frac{4 y^{3}}{3} - 4 (\frac{1}{2} y^{2}) + C$

Step 13.11

Simplify.

Tap for more steps...

Step 13.11.1

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

$g (y) = - \frac{y^{4}}{4} + \frac{4 y^{3}}{3} - 4 \frac{y^{2}}{2} + C$

Step 13.11.2

Combine $- 4$ and $\frac{y^{2}}{2}$ .

$g (y) = - \frac{y^{4}}{4} + \frac{4 y^{3}}{3} + \frac{- 4 y^{2}}{2} + C$

Step 13.11.3

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

Tap for more steps...

Step 13.11.3.1

Factor $2$ out of $- 4 y^{2}$ .

$g (y) = - \frac{y^{4}}{4} + \frac{4 y^{3}}{3} + \frac{2 (- 2 y^{2})}{2} + C$

Step 13.11.3.2

Cancel the common factors.

Tap for more steps...

Step 13.11.3.2.1

Factor $2$ out of $2$ .

$g (y) = - \frac{y^{4}}{4} + \frac{4 y^{3}}{3} + \frac{2 (- 2 y^{2})}{2 (1)} + C$

Step 13.11.3.2.2

Cancel the common factor.

$g (y) = - \frac{y^{4}}{4} + \frac{4 y^{3}}{3} + \frac{2 (- 2 y^{2})}{2 \cdot 1} + C$

Step 13.11.3.2.3

Rewrite the expression.

$g (y) = - \frac{y^{4}}{4} + \frac{4 y^{3}}{3} + \frac{- 2 y^{2}}{1} + C$

Step 13.11.3.2.4

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

$g (y) = - \frac{y^{4}}{4} + \frac{4 y^{3}}{3} - 2 y^{2} + C$

Step 13.12

Simplify.

Tap for more steps...

Step 13.12.1

Reorder terms.

$g (y) = - (\frac{1}{4} y^{4}) + \frac{4}{3} y^{3} - 2 y^{2} + C$

Step 13.12.2

Remove parentheses.

$g (y) = - \frac{1}{4} y^{4} + \frac{4}{3} y^{3} - 2 y^{2} + C$

Step 14

Substitute for $g (y)$ in $f (x, y) = (y^{3} - 6 y^{2} + 12 y - 8) x + g (y)$ .

$f (x, y) = (y^{3} - 6 y^{2} + 12 y - 8) x - \frac{1}{4} y^{4} + \frac{4}{3} y^{3} - 2 y^{2} + C$

Step 15

Simplify each term.

Tap for more steps...

Step 15.1

Apply the distributive property.

$f (x, y) = y^{3} x - 6 y^{2} x + 12 y x - 8 x - \frac{1}{4} y^{4} + \frac{4}{3} y^{3} - 2 y^{2} + C$

Step 15.2

Combine $y^{4}$ and $\frac{1}{4}$ .

$f (x, y) = y^{3} x - 6 y^{2} x + 12 y x - 8 x - \frac{y^{4}}{4} + \frac{4}{3} y^{3} - 2 y^{2} + C$

Step 15.3

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

$f (x, y) = y^{3} x - 6 y^{2} x + 12 y x - 8 x - \frac{y^{4}}{4} + \frac{4 y^{3}}{3} - 2 y^{2} + C$