Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

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

Tap for more steps...

Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Multiply $- 1$ by $1$ .

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

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $- 1$ by $1$ .

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

Step 3

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

Tap for more steps...

Step 3.1

Substitute $x^{2} - 1$ for $\frac{\partial M}{\partial y}$ and $- 1$ for $\frac{\partial N}{\partial x}$ .

$x^{2} - 1 = - 1$

Step 3.2

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

$x^{2} - 1 = - 1$ 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 $x^{2} - 1$ for $\frac{\partial M}{\partial y}$ .

$\frac{x^{2} - 1 - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $- 1$ for $\frac{\partial N}{\partial x}$ .

$\frac{x^{2} - 1 + 1}{N}$

Step 4.3

Substitute $- x$ for $N$ .

Tap for more steps...

Step 4.3.1

Substitute $- x$ for $N$ .

$\frac{x^{2} - 1 + 1}{- x}$

Step 4.3.2

Simplify the numerator.

Tap for more steps...

Step 4.3.2.1

Add $- 1$ and $1$ .

$\frac{x^{2} + 0}{- x}$

Step 4.3.2.2

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

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

Step 4.3.3

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

Tap for more steps...

Step 4.3.3.1

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

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

Step 4.3.3.2

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$- 1 \cdot x$

Step 4.3.4

Rewrite $- 1 \cdot x$ as $- x$ .

$- x$

Step 4.4

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

$μ (x, y) = e^{\int - x d x}$

Step 5

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

Tap for more steps...

Step 5.1

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

$μ (x, y) = e^{- \int x d x}$

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

Tap for more steps...

Step 8.1

Apply the constant rule.

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

Step 8.2

Rewrite $- x e^{- \frac{x^{2}}{2}} y + C$ as $- x e^{- \frac{1}{2} x^{2}} y + C$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

Tap for more steps...

Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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 $g (x) = e^{- \frac{x^{2}}{2}}$ .

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

Step 11.3.4

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

Tap for more steps...

Step 11.3.4.1

To apply the Chain Rule, set $u$ as $- \frac{x^{2}}{2}$ .

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

Step 11.3.4.2

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

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

Step 11.3.4.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{2}$ .

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

Step 11.3.5

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

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

Step 11.3.6

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

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

Step 11.3.7

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

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

Step 11.3.8

Multiply $2$ by $- 1$ .

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

Step 11.3.9

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

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

Step 11.3.10

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

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

Step 11.3.11

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

Tap for more steps...

Step 11.3.11.1

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

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

Step 11.3.11.2

Cancel the common factors.

Tap for more steps...

Step 11.3.11.2.1

Factor $2$ out of $2$ .

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

Step 11.3.11.2.2

Cancel the common factor.

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

Step 11.3.11.2.3

Rewrite the expression.

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

Step 11.3.11.2.4

Divide $- x$ by $1$ .

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

Step 11.3.12

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

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

Step 11.3.13

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

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

Step 11.3.14

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

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

Step 11.3.15

Add $1$ and $1$ .

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

Step 11.3.16

Move $- 1$ to the left of $e^{- \frac{x^{2}}{2}}$ .

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

Step 11.3.17

Rewrite $- 1 e^{- \frac{x^{2}}{2}}$ as $- e^{- \frac{x^{2}}{2}}$ .

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

Step 11.3.18

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

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

Step 11.4

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

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

Step 11.5

Simplify.

Tap for more steps...

Step 11.5.1

Apply the distributive property.

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

Step 11.5.2

Combine terms.

Tap for more steps...

Step 11.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 11.5.2.2

Multiply $y$ by $1$ .

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

Step 11.5.3

Reorder terms.

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

Step 11.5.4

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

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

Step 12

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

Tap for more steps...

Step 12.1

Move all terms not containing $\frac{d g}{d x}$ to the right side of the equation.

Tap for more steps...

Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.1.3

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

Tap for more steps...

Step 12.1.3.1

Subtract $x^{2} y e^{- \frac{x^{2}}{2}}$ from $x^{2} y e^{- \frac{x^{2}}{2}}$ .

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

Step 12.1.3.2

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

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

Step 12.1.3.3

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

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

Step 13

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

Tap for more steps...

Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

$g (x) = 0 + C$

Step 13.4

Add $0$ and $C$ .

$g (x) = C$

Step 14

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

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

Step 15

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

Tap for more steps...

Step 15.1

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

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

Step 15.2

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

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