Calculus Examples

$\frac{d y}{d x} + \frac{y}{x} = 4$

Step 1

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

Tap for more steps...

Step 1.1

Factor $y$ out of $\frac{y}{x}$ .

$\frac{d y}{d x} + y \frac{1}{x} = 4$

Step 1.2

Reorder $y$ and $\frac{1}{x}$ .

$\frac{d y}{d x} + \frac{1}{x} y = 4$

Step 2

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

Tap for more steps...

Step 2.1

Set up the integration.

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

Step 2.2

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

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

Step 2.3

Remove the constant of integration.

$e^{\ln (x)}$

Step 2.4

Exponentiation and log are inverse functions.

$x$

Step 3

Multiply each term by the integrating factor $x$ .

Tap for more steps...

Step 3.1

Multiply each term by $x$ .

$x \frac{d y}{d x} + x (\frac{1}{x} y) = x \cdot 4$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

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

$x \frac{d y}{d x} + x \frac{y}{x} = x \cdot 4$

Step 3.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.2.2.1

Cancel the common factor.

$x \frac{d y}{d x} + x \frac{y}{x} = x \cdot 4$

Step 3.2.2.2

Rewrite the expression.

$x \frac{d y}{d x} + y = x \cdot 4$

Step 3.3

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

$x \frac{d y}{d x} + y = 4 x$

Step 4

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

$\frac{d}{d x} [x y] = 4 x$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x y] d x = \int 4 x d x$

Step 6

Integrate the left side.

$x y = \int 4 x d x$

Step 7

Integrate the right side.

Tap for more steps...

Step 7.1

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

$x y = 4 \int x d x$

Step 7.2

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

$x y = 4 (\frac{1}{2} x^{2} + C)$

Step 7.3

Simplify the answer.

Tap for more steps...

Step 7.3.1

Rewrite $4 (\frac{1}{2} x^{2} + C)$ as $4 (\frac{1}{2}) x^{2} + C$ .

$x y = 4 (\frac{1}{2}) x^{2} + C$

Step 7.3.2

Simplify.

Tap for more steps...

Step 7.3.2.1

Combine $4$ and $\frac{1}{2}$ .

$x y = \frac{4}{2} x^{2} + C$

Step 7.3.2.2

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

Tap for more steps...

Step 7.3.2.2.1

Factor $2$ out of $4$ .

$x y = \frac{2 \cdot 2}{2} x^{2} + C$

Step 7.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 7.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 7.3.2.2.2.2

Cancel the common factor.

$x y = \frac{2 \cdot 2}{2 \cdot 1} x^{2} + C$

Step 7.3.2.2.2.3

Rewrite the expression.

$x y = \frac{2}{1} x^{2} + C$

Step 7.3.2.2.2.4

Divide $2$ by $1$ .

$x y = 2 x^{2} + C$

Step 8

Divide each term in $x y = 2 x^{2} + C$ by $x$ and simplify.

Tap for more steps...

Step 8.1

Divide each term in $x y = 2 x^{2} + C$ by $x$ .

$\frac{x y}{x} = \frac{2 x^{2}}{x} + \frac{C}{x}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{2 x^{2}}{x} + \frac{C}{x}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x^{2}}{x} + \frac{C}{x}$

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

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

Tap for more steps...

Step 8.3.1.1

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

$y = \frac{x (2 x)}{x} + \frac{C}{x}$

Step 8.3.1.2

Cancel the common factors.

Tap for more steps...

Step 8.3.1.2.1

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

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

Step 8.3.1.2.2

Factor $x$ out of $x^{1}$ .

$y = \frac{x (2 x)}{x \cdot 1} + \frac{C}{x}$

Step 8.3.1.2.3

Cancel the common factor.

$y = \frac{x (2 x)}{x \cdot 1} + \frac{C}{x}$

Step 8.3.1.2.4

Rewrite the expression.

$y = \frac{2 x}{1} + \frac{C}{x}$

Step 8.3.1.2.5

Divide $2 x$ by $1$ .

$y = 2 x + \frac{C}{x}$