Calculus Examples

$x \ln (x) \frac{d y}{d x} + y = x e^{x}$

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

Tap for more steps...

Step 1.1

Divide each term in $x \ln (x) \frac{d y}{d x} + y = x e^{x}$ by $x \ln (x)$ .

$\frac{x \ln (x) \frac{d y}{d x}}{x \ln (x)} + \frac{y}{x \ln (x)} = \frac{x e^{x}}{x \ln (x)}$

Step 1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.2.1

Cancel the common factor.

$\frac{x \ln (x) \frac{d y}{d x}}{x \ln (x)} + \frac{y}{x \ln (x)} = \frac{x e^{x}}{x \ln (x)}$

Step 1.2.2

Rewrite the expression.

$\frac{\ln (x) \frac{d y}{d x}}{\ln (x)} + \frac{y}{x \ln (x)} = \frac{x e^{x}}{x \ln (x)}$

Step 1.3

Cancel the common factor of $\ln (x)$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

$\frac{\ln (x) \frac{d y}{d x}}{\ln (x)} + \frac{y}{x \ln (x)} = \frac{x e^{x}}{x \ln (x)}$

Step 1.3.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} + \frac{y}{x \ln (x)} = \frac{x e^{x}}{x \ln (x)}$

Step 1.4

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

$\frac{d y}{d x} + \frac{y}{x \ln (x)} = \frac{x e^{x}}{x \ln (x)}$

Step 1.4.2

Rewrite the expression.

$\frac{d y}{d x} + \frac{y}{x \ln (x)} = \frac{e^{x}}{\ln (x)}$

Step 1.5

Factor $y$ out of $\frac{y}{x \ln (x)}$ .

$\frac{d y}{d x} + y \frac{1}{x \ln (x)} = \frac{e^{x}}{\ln (x)}$

Step 1.6

Reorder $y$ and $\frac{1}{x \ln (x)}$ .

$\frac{d y}{d x} + \frac{1}{x \ln (x)} y = \frac{e^{x}}{\ln (x)}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{x \ln (x)}$ .

Tap for more steps...

Step 2.1

Set up the integration.

$e^{\int \frac{1}{x \ln (x)} d x}$

Step 2.2

Integrate $\frac{1}{x \ln (x)}$ .

Tap for more steps...

Step 2.2.1

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.2.1.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.2.1.1.1

Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

Step 2.2.1.1.2

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

$\frac{1}{x}$

Step 2.2.1.2

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

$e^{\int \frac{1}{u} d u}$

Step 2.2.2

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

$e^{\ln (| u |) + C}$

Step 2.2.3

Replace all occurrences of $u$ with $\ln (x)$ .

$e^{\ln (| \ln (x) |) + C}$

Step 2.3

Remove the constant of integration.

$e^{\ln (\ln (x))}$

Step 2.4

Exponentiation and log are inverse functions.

$\ln (x)$

Step 3

Multiply each term by the integrating factor $\ln (x)$ .

Tap for more steps...

Step 3.1

Multiply each term by $\ln (x)$ .

$\ln (x) \frac{d y}{d x} + \ln (x) (\frac{1}{x \ln (x)} y) = \ln (x) \frac{e^{x}}{\ln (x)}$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Combine $\frac{1}{x \ln (x)}$ and $y$ .

$\ln (x) \frac{d y}{d x} + \ln (x) \frac{y}{x \ln (x)} = \ln (x) \frac{e^{x}}{\ln (x)}$

Step 3.2.2

Cancel the common factor of $\ln (x)$ .

Tap for more steps...

Step 3.2.2.1

Factor $\ln (x)$ out of $x \ln (x)$ .

$\ln (x) \frac{d y}{d x} + \ln (x) \frac{y}{\ln (x) x} = \ln (x) \frac{e^{x}}{\ln (x)}$

Step 3.2.2.2

Cancel the common factor.

$\ln (x) \frac{d y}{d x} + \ln (x) \frac{y}{\ln (x) x} = \ln (x) \frac{e^{x}}{\ln (x)}$

Step 3.2.2.3

Rewrite the expression.

$\ln (x) \frac{d y}{d x} + \frac{y}{x} = \ln (x) \frac{e^{x}}{\ln (x)}$

Step 3.3

Cancel the common factor of $\ln (x)$ .

Tap for more steps...

Step 3.3.1

Cancel the common factor.

$\ln (x) \frac{d y}{d x} + \frac{y}{x} = \ln (x) \frac{e^{x}}{\ln (x)}$

Step 3.3.2

Rewrite the expression.

$\ln (x) \frac{d y}{d x} + \frac{y}{x} = e^{x}$

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [\ln (x) y] = e^{x}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\ln (x) y] d x = \int e^{x} d x$

Step 6

Integrate the left side.

$\ln (x) y = \int e^{x} d x$

Step 7

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

$\ln (x) y = e^{x} + C$

Step 8

Divide each term in $\ln (x) y = e^{x} + C$ by $\ln (x)$ and simplify.

Tap for more steps...

Step 8.1

Divide each term in $\ln (x) y = e^{x} + C$ by $\ln (x)$ .

$\frac{\ln (x) y}{\ln (x)} = \frac{e^{x}}{\ln (x)} + \frac{C}{\ln (x)}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor of $\ln (x)$ .

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{\ln (x) y}{\ln (x)} = \frac{e^{x}}{\ln (x)} + \frac{C}{\ln (x)}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{e^{x}}{\ln (x)} + \frac{C}{\ln (x)}$