Calculus Examples

$x \frac{d y}{d x} + 5 y = \frac{\ln (x)}{x}$

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 $x \frac{d y}{d x} + 5 y = \frac{\ln (x)}{x}$ by $x$ .

$\frac{x \frac{d y}{d x}}{x} + \frac{5 y}{x} = \frac{\frac{\ln (x)}{x}}{x}$

Step 1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.2.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} + \frac{5 y}{x} = \frac{\frac{\ln (x)}{x}}{x}$

Step 1.2.2

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

$\frac{d y}{d x} + \frac{5 y}{x} = \frac{\frac{\ln (x)}{x}}{x}$

Step 1.3

Factor $y$ out of $\frac{5 y}{x}$ .

$\frac{d y}{d x} + y \frac{5}{x} = \frac{\frac{\ln (x)}{x}}{x}$

Step 1.4

Reorder $y$ and $\frac{5}{x}$ .

$\frac{d y}{d x} + \frac{5}{x} y = \frac{\frac{\ln (x)}{x}}{x}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{5}{x}$ .

Tap for more steps...

Step 2.1

Set up the integration.

$e^{\int \frac{5}{x} d x}$

Step 2.2

Integrate $\frac{5}{x}$ .

Tap for more steps...

Step 2.2.1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$e^{5 \int \frac{1}{x} d x}$

Step 2.2.2

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

$e^{5 (\ln (| x |) + C)}$

Step 2.2.3

Simplify.

$e^{5 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{5 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{5})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{5}$

Step 3

Multiply each term by the integrating factor $x^{5}$ .

Tap for more steps...

Step 3.1

Multiply each term by $x^{5}$ .

$x^{5} \frac{d y}{d x} + x^{5} (\frac{5}{x} y) = x^{5} \frac{\frac{\ln (x)}{x}}{x}$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Combine $\frac{5}{x}$ and $y$ .

$x^{5} \frac{d y}{d x} + x^{5} \frac{5 y}{x} = x^{5} \frac{\frac{\ln (x)}{x}}{x}$

Step 3.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.2.2.1

Factor $x$ out of $x^{5}$ .

$x^{5} \frac{d y}{d x} + x \cdot x^{4} \frac{5 y}{x} = x^{5} \frac{\frac{\ln (x)}{x}}{x}$

Step 3.2.2.2

Cancel the common factor.

$x^{5} \frac{d y}{d x} + x \cdot x^{4} \frac{5 y}{x} = x^{5} \frac{\frac{\ln (x)}{x}}{x}$

Step 3.2.2.3

Rewrite the expression.

$x^{5} \frac{d y}{d x} + x^{4} (5 y) = x^{5} \frac{\frac{\ln (x)}{x}}{x}$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$x^{5} \frac{d y}{d x} + 5 x^{4} y = x^{5} \frac{\frac{\ln (x)}{x}}{x}$

Step 3.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.3.1

Factor $x$ out of $x^{5}$ .

$x^{5} \frac{d y}{d x} + 5 x^{4} y = x \cdot x^{4} \frac{\frac{\ln (x)}{x}}{x}$

Step 3.3.2

Cancel the common factor.

$x^{5} \frac{d y}{d x} + 5 x^{4} y = x \cdot x^{4} \frac{\frac{\ln (x)}{x}}{x}$

Step 3.3.3

Rewrite the expression.

$x^{5} \frac{d y}{d x} + 5 x^{4} y = x^{4} \frac{\ln (x)}{x}$

Step 3.4

Combine $x^{4}$ and $\frac{\ln (x)}{x}$ .

$x^{5} \frac{d y}{d x} + 5 x^{4} y = \frac{x^{4} \ln (x)}{x}$

Step 3.5

Cancel the common factor of $x^{4}$ and $x$ .

Tap for more steps...

Step 3.5.1

Factor $x$ out of $x^{4} \ln (x)$ .

$x^{5} \frac{d y}{d x} + 5 x^{4} y = \frac{x (x^{3} \ln (x))}{x}$

Step 3.5.2

Cancel the common factors.

Tap for more steps...

Step 3.5.2.1

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

$x^{5} \frac{d y}{d x} + 5 x^{4} y = \frac{x (x^{3} \ln (x))}{x^{1}}$

Step 3.5.2.2

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

$x^{5} \frac{d y}{d x} + 5 x^{4} y = \frac{x (x^{3} \ln (x))}{x \cdot 1}$

Step 3.5.2.3

Cancel the common factor.

$x^{5} \frac{d y}{d x} + 5 x^{4} y = \frac{x (x^{3} \ln (x))}{x \cdot 1}$

Step 3.5.2.4

Rewrite the expression.

$x^{5} \frac{d y}{d x} + 5 x^{4} y = \frac{x^{3} \ln (x)}{1}$

Step 3.5.2.5

Divide $x^{3} \ln (x)$ by $1$ .

$x^{5} \frac{d y}{d x} + 5 x^{4} y = x^{3} \ln (x)$

Step 4

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

$\frac{d}{d x} [x^{5} y] = x^{3} \ln (x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{5} y] d x = \int x^{3} \ln (x) d x$

Step 6

Integrate the left side.

$x^{5} y = \int x^{3} \ln (x) d x$

Step 7

Integrate the right side.

Tap for more steps...

Step 7.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{3}$ .

$x^{5} y = \ln (x) (\frac{1}{4} x^{4}) - \int \frac{1}{4} x^{4} \frac{1}{x} d x$

Step 7.2

Simplify.

Tap for more steps...

Step 7.2.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$x^{5} y = \ln (x) \frac{x^{4}}{4} - \int \frac{1}{4} x^{4} \frac{1}{x} d x$

Step 7.2.2

Combine $\ln (x)$ and $\frac{x^{4}}{4}$ .

$x^{5} y = \frac{\ln (x) x^{4}}{4} - \int \frac{1}{4} x^{4} \frac{1}{x} d x$

Step 7.3

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$x^{5} y = \frac{\ln (x) x^{4}}{4} - (\frac{1}{4} \int x^{4} \frac{1}{x} d x)$

Step 7.4

Simplify.

Tap for more steps...

Step 7.4.1

Combine $x^{4}$ and $\frac{1}{x}$ .

$x^{5} y = \frac{\ln (x) x^{4}}{4} - (\frac{1}{4} \int \frac{x^{4}}{x} d x)$

Step 7.4.2

Cancel the common factor of $x^{4}$ and $x$ .

Tap for more steps...

Step 7.4.2.1

Factor $x$ out of $x^{4}$ .

$x^{5} y = \frac{\ln (x) x^{4}}{4} - (\frac{1}{4} \int \frac{x \cdot x^{3}}{x} d x)$

Step 7.4.2.2

Cancel the common factors.

Tap for more steps...

Step 7.4.2.2.1

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

$x^{5} y = \frac{\ln (x) x^{4}}{4} - (\frac{1}{4} \int \frac{x \cdot x^{3}}{x^{1}} d x)$

Step 7.4.2.2.2

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

$x^{5} y = \frac{\ln (x) x^{4}}{4} - (\frac{1}{4} \int \frac{x \cdot x^{3}}{x \cdot 1} d x)$

Step 7.4.2.2.3

Cancel the common factor.

$x^{5} y = \frac{\ln (x) x^{4}}{4} - (\frac{1}{4} \int \frac{x \cdot x^{3}}{x \cdot 1} d x)$

Step 7.4.2.2.4

Rewrite the expression.

$x^{5} y = \frac{\ln (x) x^{4}}{4} - (\frac{1}{4} \int \frac{x^{3}}{1} d x)$

Step 7.4.2.2.5

Divide $x^{3}$ by $1$ .

$x^{5} y = \frac{\ln (x) x^{4}}{4} - (\frac{1}{4} \int x^{3} d x)$

$x^{5} y = \frac{\ln (x) x^{4}}{4} - \frac{1}{4} \int x^{3} d x$

Step 7.5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$x^{5} y = \frac{\ln (x) x^{4}}{4} - \frac{1}{4} (\frac{1}{4} x^{4} + C)$

Step 7.6

Simplify the answer.

Tap for more steps...

Step 7.6.1

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

$x^{5} y = \frac{1}{4} \ln (x) x^{4} - \frac{1}{4} \cdot \frac{1}{4} x^{4} + C$

Step 7.6.2

Simplify.

Tap for more steps...

Step 7.6.2.1

Combine $\frac{1}{4}$ and $\ln (x)$ .

$x^{5} y = \frac{\ln (x)}{4} x^{4} - \frac{1}{4} \cdot \frac{1}{4} x^{4} + C$

Step 7.6.2.2

Combine $\frac{\ln (x)}{4}$ and $x^{4}$ .

$x^{5} y = \frac{\ln (x) x^{4}}{4} - \frac{1}{4} \cdot \frac{1}{4} x^{4} + C$

Step 7.6.2.3

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

$x^{5} y = \frac{\ln (x) x^{4}}{4} - \frac{1}{4 \cdot 4} x^{4} + C$

Step 7.6.2.4

Multiply $4$ by $4$ .

$x^{5} y = \frac{\ln (x) x^{4}}{4} - \frac{1}{16} x^{4} + C$

$x^{5} y = \frac{1}{4} \ln (x) x^{4} - \frac{1}{16} x^{4} + C$

Step 8

Solve for $y$ .

Tap for more steps...

Step 8.1

Simplify.

Tap for more steps...

Step 8.1.1

Combine $x^{4}$ and $\frac{1}{16}$ .

$x^{5} y = \frac{1}{4} \cdot (\ln (x) x^{4}) - \frac{x^{4}}{16} + C$

Step 8.1.2

Remove parentheses.

$x^{5} y = \frac{1}{4} \cdot \ln (x) x^{4} - \frac{x^{4}}{16} + C$

Step 8.2

Divide each term in $x^{5} y = \frac{1}{4} \cdot (\ln (x) x^{4}) - \frac{x^{4}}{16} + C$ by $x^{5}$ and simplify.

Tap for more steps...

Step 8.2.1

Divide each term in $x^{5} y = \frac{1}{4} \cdot (\ln (x) x^{4}) - \frac{x^{4}}{16} + C$ by $x^{5}$ .

$\frac{x^{5} y}{x^{5}} = \frac{\frac{1}{4} \cdot (\ln (x) x^{4})}{x^{5}} + \frac{- \frac{x^{4}}{16}}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.2

Simplify the left side.

Tap for more steps...

Step 8.2.2.1

Cancel the common factor of $x^{5}$ .

Tap for more steps...

Step 8.2.2.1.1

Cancel the common factor.

$\frac{x^{5} y}{x^{5}} = \frac{\frac{1}{4} \cdot (\ln (x) x^{4})}{x^{5}} + \frac{- \frac{x^{4}}{16}}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{4} \cdot (\ln (x) x^{4})}{x^{5}} + \frac{- \frac{x^{4}}{16}}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.3

Simplify the right side.

Tap for more steps...

Step 8.2.3.1

Simplify each term.

Tap for more steps...

Step 8.2.3.1.1

Cancel the common factor of $x^{4}$ and $x^{5}$ .

Tap for more steps...

Step 8.2.3.1.1.1

Factor $x^{4}$ out of $\frac{1}{4} \cdot (\ln (x) x^{4})$ .

$y = \frac{x^{4} (\frac{1}{4} \cdot (\ln (x)))}{x^{5}} + \frac{- \frac{x^{4}}{16}}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 8.2.3.1.1.2.1

Factor $x^{4}$ out of $x^{5}$ .

$y = \frac{x^{4} (\frac{1}{4} \cdot (\ln (x)))}{x^{4} x} + \frac{- \frac{x^{4}}{16}}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{4} (\frac{1}{4} \cdot (\ln (x)))}{x^{4} x} + \frac{- \frac{x^{4}}{16}}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{4} \cdot (\ln (x))}{x} + \frac{- \frac{x^{4}}{16}}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.3.1.2

Simplify $\frac{1}{4} \ln (x)$ by moving $\frac{1}{4}$ inside the logarithm.

$y = \frac{\ln (x^{\frac{1}{4}})}{x} + \frac{- \frac{x^{4}}{16}}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{\ln (x^{\frac{1}{4}})}{x} - \frac{x^{4}}{16} \cdot \frac{1}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.3.1.4

Cancel the common factor of $x^{4}$ .

Tap for more steps...

Step 8.2.3.1.4.1

Move the leading negative in $- \frac{x^{4}}{16}$ into the numerator.

$y = \frac{\ln (x^{\frac{1}{4}})}{x} + \frac{- x^{4}}{16} \cdot \frac{1}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.3.1.4.2

Factor $x^{4}$ out of $- x^{4}$ .

$y = \frac{\ln (x^{\frac{1}{4}})}{x} + \frac{x^{4} \cdot - 1}{16} \cdot \frac{1}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.3.1.4.3

Factor $x^{4}$ out of $x^{5}$ .

$y = \frac{\ln (x^{\frac{1}{4}})}{x} + \frac{x^{4} \cdot - 1}{16} \cdot \frac{1}{x^{4} x} + \frac{C}{x^{5}}$

Step 8.2.3.1.4.4

Cancel the common factor.

$y = \frac{\ln (x^{\frac{1}{4}})}{x} + \frac{x^{4} \cdot - 1}{16} \cdot \frac{1}{x^{4} x} + \frac{C}{x^{5}}$

Step 8.2.3.1.4.5

Rewrite the expression.

$y = \frac{\ln (x^{\frac{1}{4}})}{x} + \frac{- 1}{16} \cdot \frac{1}{x} + \frac{C}{x^{5}}$

Step 8.2.3.1.5

Multiply $\frac{- 1}{16}$ by $\frac{1}{x}$ .

$y = \frac{\ln (x^{\frac{1}{4}})}{x} + \frac{- 1}{16 x} + \frac{C}{x^{5}}$

Step 8.2.3.1.6

Move the negative in front of the fraction.

$y = \frac{\ln (x^{\frac{1}{4}})}{x} - \frac{1}{16 x} + \frac{C}{x^{5}}$