Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 0 + 2 \cdot 1$

Step 1.3.3

Multiply $2$ by $1$ .

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

Step 1.4

Add $0$ and $2$ .

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

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

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

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

Step 3

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

Tap for more steps...

Step 3.1

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

$2 = 1$

Step 3.2

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

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

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

Step 4.2

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

$\frac{2 - 1}{N}$

Step 4.3

Substitute $x$ for $N$ .

Tap for more steps...

Step 4.3.1

Substitute $x$ for $N$ .

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

Step 4.3.2

Subtract $1$ from $2$ .

$\frac{1}{x}$

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 $(x + 2 y) d x + x d y = 0$ by the integration factor $x$ .

Tap for more steps...

Step 6.1

Multiply $x + 2 y$ by $x$ .

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

Step 6.2

Apply the distributive property.

$(x \cdot x + 2 y x) d x + x d y = 0$

Step 6.3

Multiply $x$ by $x$ .

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

Step 6.4

Multiply $x$ by $x$ .

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

Step 6.5

Multiply $x$ by $x$ .

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

Step 7

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

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

Step 8

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

Tap for more steps...

Step 8.1

Apply the constant rule.

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

Step 9

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

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

Step 10

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

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

Step 11

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

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

Tap for more steps...

Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

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 $2 x y$ from both sides of the equation.

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

Step 12.1.2

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

Tap for more steps...

Step 12.1.2.1

Reorder the factors in the terms $2 y x$ and $- 2 x y$ .

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

Step 12.1.2.2

Subtract $2 x y$ from $2 x y$ .

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

Step 12.1.2.3

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

$\frac{d g}{d x} = x^{2}$

Step 13

Find the antiderivative of $x^{2}$ to find $g (x)$ .

Tap for more steps...

Step 13.1

Integrate both sides of $\frac{d g}{d x} = x^{2}$ .

$\int \frac{d g}{d x} d x = \int x^{2} d x$

Step 13.2

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

$g (x) = \int x^{2} d x$

Step 13.3

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

$g (x) = \frac{1}{3} x^{3} + C$

Step 14

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

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

Step 15

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

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