Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

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

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

$\frac{\partial N}{\partial x} = - 2 (1 + 0)$

Step 2.6

Simplify the expression.

Tap for more steps...

Step 2.6.1

Add $1$ and $0$ .

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

Step 2.6.2

Multiply $- 2$ by $1$ .

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

Step 3

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

Tap for more steps...

Step 3.1

Substitute $0$ for $\frac{\partial M}{\partial y}$ and $- 2$ for $\frac{\partial N}{\partial x}$ .

$0 = - 2$

Step 3.2

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

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

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

Step 4.2

Substitute $0$ for $\frac{\partial M}{\partial y}$ .

$\frac{- 2 + 0}{M}$

Step 4.3

Substitute $1$ for $M$ .

Tap for more steps...

Step 4.3.1

Substitute $1$ for $M$ .

$\frac{- 2 + 0}{1}$

Step 4.3.2

Divide $- 2 + 0$ by $1$ .

$- 2 + 0$

Step 4.3.3

Substitute $1$ for $M$ .

$- 2$

Step 4.4

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

$μ (x, y) = e^{\int - 2 d y}$

Step 5

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

Tap for more steps...

Step 5.1

Apply the constant rule.

$μ (x, y) = e^{- 2 y + C}$

Step 5.2

Simplify.

$μ (x, y) = e^{- 2 y} + C$

Step 6

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

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

Step 7

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

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

Step 8

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

Tap for more steps...

Step 8.1

Apply the constant rule.

$f (x, y) = e^{- 2 y} x + C$

Step 9

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

$f (x, y) = e^{- 2 y} x + g (y)$

Step 10

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

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

Step 11

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

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

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

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

Step 11.3

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

Tap for more steps...

Step 11.3.1

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

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

Step 11.3.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = e^{y}$ and $g (y) = - 2 y$ .

Tap for more steps...

Step 11.3.2.1

To apply the Chain Rule, set $u$ as $- 2 y$ .

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

Step 11.3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 11.3.2.3

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

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

Step 11.3.3

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

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

Step 11.3.4

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

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

Step 11.3.5

Multiply $- 2$ by $1$ .

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

Step 11.3.6

Move $- 2$ to the left of $e^{- 2 y}$ .

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

Step 11.4

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

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

Step 11.5

Simplify.

Tap for more steps...

Step 11.5.1

Reorder terms.

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

Step 11.5.2

Reorder factors in $\frac{d g}{d y} - 2 e^{- 2 y} x$ .

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

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

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

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

Step 12.1.2

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

Tap for more steps...

Step 12.1.2.1

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

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

Step 12.1.2.2

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

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

Step 13

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

Tap for more steps...

Step 13.1

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

$\int \frac{d g}{d y} d y = \int - 2 y e^{- 2 y} d y$

Step 13.2

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

$g (y) = \int - 2 y e^{- 2 y} d y$

Step 13.3

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

$g (y) = - 2 \int y e^{- 2 y} d y$

Step 13.4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = y$ and $d v = e^{- 2 y}$ .

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

Step 13.5

Simplify.

Tap for more steps...

Step 13.5.1

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

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

Step 13.5.2

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

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

Step 13.5.3

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

$g (y) = - 2 (- \frac{y e^{- 2 y}}{2} - \int - \frac{e^{- 2 y}}{2} d y)$

Step 13.6

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

$g (y) = - 2 (- \frac{y e^{- 2 y}}{2} - - \int \frac{e^{- 2 y}}{2} d y)$

Step 13.7

Simplify.

Tap for more steps...

Step 13.7.1

Multiply $- 1$ by $- 1$ .

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

Step 13.7.2

Multiply $\int \frac{e^{- 2 y}}{2} d y$ by $1$ .

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

Step 13.8

Since $\frac{1}{2}$ is constant with respect to $y$ , move $\frac{1}{2}$ out of the integral.

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

Step 13.9

Let $u = - 2 y$ . Then $d u = - 2 d y$ , so $- \frac{1}{2} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 13.9.1

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

Tap for more steps...

Step 13.9.1.1

Differentiate $- 2 y$ .

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

Step 13.9.1.2

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

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

Step 13.9.1.3

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

$- 2 \cdot 1$

Step 13.9.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 13.9.2

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

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

Step 13.10

Simplify.

Tap for more steps...

Step 13.10.1

Move the negative in front of the fraction.

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

Step 13.10.2

Combine $e^{u}$ and $\frac{1}{2}$ .

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

Step 13.11

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

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

Step 13.12

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 13.13

Simplify.

Tap for more steps...

Step 13.13.1

Remove parentheses.

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

Step 13.13.2

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

$g (y) = - 2 (- \frac{y e^{- 2 y}}{2} - \frac{1}{2 \cdot 2} \int e^{u} d u)$

Step 13.13.3

Multiply $2$ by $2$ .

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

Step 13.14

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

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

Step 13.15

Rewrite $- 2 (- \frac{y e^{- 2 y}}{2} - \frac{1}{4} (e^{u} + C))$ as $- 2 (- \frac{1}{2} y e^{- 2 y} - \frac{1}{4} e^{u}) + C$ .

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

Step 13.16

Simplify.

Tap for more steps...

Step 13.16.1

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

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

Step 13.16.2

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

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

Step 13.16.3

Combine $e^{u}$ and $\frac{1}{4}$ .

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

Step 13.17

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

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

Step 13.18

Simplify.

Tap for more steps...

Step 13.18.1

Apply the distributive property.

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

Step 13.18.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 13.18.2.1

Move the leading negative in $- \frac{e^{- 2 y} y}{2}$ into the numerator.

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

Step 13.18.2.2

Factor $2$ out of $- 2$ .

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

Step 13.18.2.3

Cancel the common factor.

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

Step 13.18.2.4

Rewrite the expression.

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

Step 13.18.3

Multiply $- 1$ by $- 1$ .

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

Step 13.18.4

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

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

Step 13.18.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 13.18.5.1

Move the leading negative in $- \frac{e^{- 2 y}}{4}$ into the numerator.

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

Step 13.18.5.2

Factor $2$ out of $- 2$ .

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

Step 13.18.5.3

Factor $2$ out of $4$ .

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

Step 13.18.5.4

Cancel the common factor.

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

Step 13.18.5.5

Rewrite the expression.

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

Step 13.18.6

Simplify each term.

Tap for more steps...

Step 13.18.6.1

Move the negative in front of the fraction.

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

Step 13.18.6.2

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

Tap for more steps...

Step 13.18.6.2.1

Multiply $- 1$ by $- 1$ .

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

Step 13.18.6.2.2

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

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

Step 13.18.7

Combine $e^{- 2 y} y$ and $\frac{e^{- 2 y}}{2}$ using a common denominator.

Tap for more steps...

Step 13.18.7.1

Reorder $e^{- 2 y}$ and $y$ .

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

Step 13.18.7.2

To write $y e^{- 2 y}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 13.18.7.3

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

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

Step 13.18.7.4

Combine the numerators over the common denominator.

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

Step 13.18.8

Simplify the numerator.

Tap for more steps...

Step 13.18.8.1

Factor $e^{- 2 y}$ out of $y e^{- 2 y} \cdot 2 + e^{- 2 y}$ .

Tap for more steps...

Step 13.18.8.1.1

Factor $e^{- 2 y}$ out of $y e^{- 2 y} \cdot 2$ .

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

Step 13.18.8.1.2

Multiply by $1$ .

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

Step 13.18.8.1.3

Factor $e^{- 2 y}$ out of $e^{- 2 y} (y \cdot 2) + e^{- 2 y} \cdot 1$ .

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

Step 13.18.8.2

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

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

Step 13.19

Reorder terms.

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

Step 14

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

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

Step 15

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

Tap for more steps...

Step 15.1

Simplify each term.

Tap for more steps...

Step 15.1.1

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

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

Step 15.1.2

Apply the distributive property.

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

Step 15.1.3

Rewrite using the commutative property of multiplication.

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

Step 15.1.4

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

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

Step 15.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 15.1.5.1

Cancel the common factor.

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

Step 15.1.5.2

Rewrite the expression.

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

Step 15.1.6

Combine $e^{- 2 y} y$ and $\frac{e^{- 2 y}}{2}$ using a common denominator.

Tap for more steps...

Step 15.1.6.1

Reorder $e^{- 2 y}$ and $y$ .

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

Step 15.1.6.2

To write $y e^{- 2 y}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 15.1.6.3

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

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

Step 15.1.6.4

Combine the numerators over the common denominator.

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

Step 15.1.7

Simplify the numerator.

Tap for more steps...

Step 15.1.7.1

Factor $e^{- 2 y}$ out of $y e^{- 2 y} \cdot 2 + e^{- 2 y}$ .

Tap for more steps...

Step 15.1.7.1.1

Factor $e^{- 2 y}$ out of $y e^{- 2 y} \cdot 2$ .

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

Step 15.1.7.1.2

Multiply by $1$ .

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

Step 15.1.7.1.3

Factor $e^{- 2 y}$ out of $e^{- 2 y} (y \cdot 2) + e^{- 2 y} \cdot 1$ .

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

Step 15.1.7.2

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

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

Step 15.2

To write $e^{- 2 y} x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 15.3

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

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

Step 15.4

Combine the numerators over the common denominator.

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

Step 15.5

Simplify the numerator.

Tap for more steps...

Step 15.5.1

Factor $e^{- 2 y}$ out of $e^{- 2 y} x \cdot 2 + e^{- 2 y} (2 y + 1)$ .

Tap for more steps...

Step 15.5.1.1

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

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

Step 15.5.1.2

Factor $e^{- 2 y}$ out of $e^{- 2 y} (x \cdot 2) + e^{- 2 y} (2 y + 1)$ .

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

Step 15.5.2

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

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