Calculus Examples

$\frac{d y}{d x} + \frac{11}{x} y = 3 x^{2}$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{11}{x}$ .

Tap for more steps...

Step 1.1

Set up the integration.

$e^{\int \frac{11}{x} d x}$

Step 1.2

Integrate $\frac{11}{x}$ .

Tap for more steps...

Step 1.2.1

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

$e^{11 \int \frac{1}{x} d x}$

Step 1.2.2

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

$e^{11 (\ln (| x |) + C)}$

Step 1.2.3

Simplify.

$e^{11 \ln (| x |) + C}$

Step 1.3

Remove the constant of integration.

$e^{11 \ln (x)}$

Step 1.4

Use the logarithmic power rule.

$e^{\ln (x^{11})}$

Step 1.5

Exponentiation and log are inverse functions.

$x^{11}$

Step 2

Multiply each term by the integrating factor $x^{11}$ .

Tap for more steps...

Step 2.1

Multiply each term by $x^{11}$ .

$x^{11} \frac{d y}{d x} + x^{11} (\frac{11}{x} y) = x^{11} (3 x^{2})$

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

Combine $\frac{11}{x}$ and $y$ .

$x^{11} \frac{d y}{d x} + x^{11} \frac{11 y}{x} = x^{11} (3 x^{2})$

Step 2.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.2.1

Factor $x$ out of $x^{11}$ .

$x^{11} \frac{d y}{d x} + x \cdot x^{10} \frac{11 y}{x} = x^{11} (3 x^{2})$

Step 2.2.2.2

Cancel the common factor.

$x^{11} \frac{d y}{d x} + x \cdot x^{10} \frac{11 y}{x} = x^{11} (3 x^{2})$

Step 2.2.2.3

Rewrite the expression.

$x^{11} \frac{d y}{d x} + x^{10} (11 y) = x^{11} (3 x^{2})$

Step 2.2.3

Rewrite using the commutative property of multiplication.

$x^{11} \frac{d y}{d x} + 11 x^{10} y = x^{11} (3 x^{2})$

Step 2.3

Rewrite using the commutative property of multiplication.

$x^{11} \frac{d y}{d x} + 11 x^{10} y = 3 x^{11} x^{2}$

Step 2.4

Multiply $x^{11}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 2.4.1

Move $x^{2}$ .

$x^{11} \frac{d y}{d x} + 11 x^{10} y = 3 (x^{2} x^{11})$

Step 2.4.2

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

$x^{11} \frac{d y}{d x} + 11 x^{10} y = 3 x^{2 + 11}$

Step 2.4.3

Add $2$ and $11$ .

$x^{11} \frac{d y}{d x} + 11 x^{10} y = 3 x^{13}$

Step 3

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

$\frac{d}{d x} [x^{11} y] = 3 x^{13}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [x^{11} y] d x = \int 3 x^{13} d x$

Step 5

Integrate the left side.

$x^{11} y = \int 3 x^{13} d x$

Step 6

Integrate the right side.

Tap for more steps...

Step 6.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$x^{11} y = 3 \int x^{13} d x$

Step 6.2

By the Power Rule, the integral of $x^{13}$ with respect to $x$ is $\frac{1}{14} x^{14}$ .

$x^{11} y = 3 (\frac{1}{14} x^{14} + C)$

Step 6.3

Simplify the answer.

Tap for more steps...

Step 6.3.1

Rewrite $3 (\frac{1}{14} x^{14} + C)$ as $3 (\frac{1}{14}) x^{14} + C$ .

$x^{11} y = 3 (\frac{1}{14}) x^{14} + C$

Step 6.3.2

Combine $3$ and $\frac{1}{14}$ .

$x^{11} y = \frac{3}{14} x^{14} + C$

Step 7

Divide each term in $x^{11} y = \frac{3}{14} x^{14} + C$ by $x^{11}$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $x^{11} y = \frac{3}{14} x^{14} + C$ by $x^{11}$ .

$\frac{x^{11} y}{x^{11}} = \frac{\frac{3}{14} x^{14}}{x^{11}} + \frac{C}{x^{11}}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $x^{11}$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{x^{11} y}{x^{11}} = \frac{\frac{3}{14} x^{14}}{x^{11}} + \frac{C}{x^{11}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{3}{14} x^{14}}{x^{11}} + \frac{C}{x^{11}}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Simplify each term.

Tap for more steps...

Step 7.3.1.1

Cancel the common factor of $x^{14}$ and $x^{11}$ .

Tap for more steps...

Step 7.3.1.1.1

Factor $x^{11}$ out of $\frac{3}{14} x^{14}$ .

$y = \frac{x^{11} (\frac{3}{14} x^{3})}{x^{11}} + \frac{C}{x^{11}}$

Step 7.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 7.3.1.1.2.1

Multiply by $1$ .

$y = \frac{x^{11} (\frac{3}{14} x^{3})}{x^{11} \cdot 1} + \frac{C}{x^{11}}$

Step 7.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{11} (\frac{3}{14} x^{3})}{x^{11} \cdot 1} + \frac{C}{x^{11}}$

Step 7.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{3}{14} x^{3}}{1} + \frac{C}{x^{11}}$

Step 7.3.1.1.2.4

Divide $\frac{3}{14} x^{3}$ by $1$ .

$y = \frac{3}{14} x^{3} + \frac{C}{x^{11}}$

Step 7.3.1.2

Combine $\frac{3}{14}$ and $x^{3}$ .

$y = \frac{3 x^{3}}{14} + \frac{C}{x^{11}}$