Calculus Examples

$x \frac{d y}{d x} - 4 y = 5 x^{4} e^{x}$

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 $x \frac{d y}{d x} - 4 y = 5 x^{4} e^{x}$ by $x$ .

$\frac{x \frac{d y}{d x}}{x} + \frac{- 4 y}{x} = \frac{5 x^{4} e^{x}}{x}$

Step 1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.2.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} + \frac{- 4 y}{x} = \frac{5 x^{4} e^{x}}{x}$

Step 1.2.2

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

$\frac{d y}{d x} + \frac{- 4 y}{x} = \frac{5 x^{4} e^{x}}{x}$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

Cancel the common factors.

Tap for more steps...

Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Cancel the common factor.

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

Step 1.3.2.4

Rewrite the expression.

$\frac{d y}{d x} + \frac{- 4 y}{x} = \frac{5 x^{3} e^{x}}{1}$

Step 1.3.2.5

Divide $5 x^{3} e^{x}$ by $1$ .

$\frac{d y}{d x} + \frac{- 4 y}{x} = 5 x^{3} e^{x}$

Step 1.4

Factor $y$ out of $\frac{- 4 y}{x}$ .

$\frac{d y}{d x} + y \frac{- 4}{x} = 5 x^{3} e^{x}$

Step 1.5

Reorder $y$ and $\frac{- 4}{x}$ .

$\frac{d y}{d x} + \frac{- 4}{x} y = 5 x^{3} e^{x}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 4}{x}$ .

Tap for more steps...

Step 2.1

Set up the integration.

$e^{\int \frac{- 4}{x} d x}$

Step 2.2

Integrate $\frac{- 4}{x}$ .

Tap for more steps...

Step 2.2.1

Split the fraction into multiple fractions.

$e^{\int - \frac{4}{x} d x}$

Step 2.2.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$e^{- \int \frac{4}{x} d x}$

Step 2.2.3

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

$e^{- (4 \int \frac{1}{x} d x)}$

Step 2.2.4

Multiply $4$ by $- 1$ .

$e^{- 4 \int \frac{1}{x} d x}$

Step 2.2.5

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

$e^{- 4 (\ln (| x |) + C)}$

Step 2.2.6

Simplify.

$e^{- 4 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- 4 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 4})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 4}$

Step 2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{4}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{4}}$ .

Tap for more steps...

Step 3.1

Multiply each term by $\frac{1}{x^{4}}$ .

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

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

Combine $\frac{4}{x}$ and $y$ .

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

Step 3.2.5

Multiply $- \frac{1}{x^{4}} \cdot \frac{4 y}{x}$ .

Tap for more steps...

Step 3.2.5.1

Multiply $\frac{4 y}{x}$ by $\frac{1}{x^{4}}$ .

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

Step 3.2.5.2

Multiply $x$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 3.2.5.2.1

Multiply $x$ by $x^{4}$ .

Tap for more steps...

Step 3.2.5.2.1.1

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

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

Step 3.2.5.2.1.2

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

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

Step 3.2.5.2.2

Add $1$ and $4$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Combine $5$ and $\frac{1}{x^{4}}$ .

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

Step 3.5

Cancel the common factor of $x^{3}$ .

Tap for more steps...

Step 3.5.1

Factor $x^{3}$ out of $x^{4}$ .

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

Step 3.5.2

Factor $x^{3}$ out of $x^{3} e^{x}$ .

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

Step 3.5.3

Cancel the common factor.

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

Step 3.5.4

Rewrite the expression.

$\frac{\frac{d y}{d x}}{x^{4}} - \frac{4 y}{x^{5}} = \frac{5}{x} e^{x}$

Step 3.6

Combine $\frac{5}{x}$ and $e^{x}$ .

$\frac{\frac{d y}{d x}}{x^{4}} - \frac{4 y}{x^{5}} = \frac{5 e^{x}}{x}$

Step 4

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

$\frac{d}{d x} [\frac{1}{x^{4}} y] = \frac{5 e^{x}}{x}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{4}} y] d x = \int \frac{5 e^{x}}{x} d x$

Step 6

Integrate the left side.

$\frac{1}{x^{4}} y = \int \frac{5 e^{x}}{x} d x$

Step 7

Integrate the right side.

Tap for more steps...

Step 7.1

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

$\frac{1}{x^{4}} y = 5 \int \frac{e^{x}}{x} d x$

Step 7.2

$\int \frac{e^{x}}{x} d x$ is a special integral. The integral is the exponential integral function.

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

Step 7.3

Simplify.

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

Step 8

Solve for $y$ .

Tap for more steps...

Step 8.1

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

$\frac{y}{x^{4}} = 5 E i x + C$

Step 8.2

Multiply both sides by $x^{4}$ .

$\frac{y}{x^{4}} x^{4} = (5 E i x + C) x^{4}$

Step 8.3

Simplify.

Tap for more steps...

Step 8.3.1

Simplify the left side.

Tap for more steps...

Step 8.3.1.1

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

Tap for more steps...

Step 8.3.1.1.1

Cancel the common factor.

$\frac{y}{x^{4}} x^{4} = (5 E i x + C) x^{4}$

Step 8.3.1.1.2

Rewrite the expression.

$y = (5 E i x + C) x^{4}$

Step 8.3.2

Simplify the right side.

Tap for more steps...

Step 8.3.2.1

Simplify $(5 E i x + C) x^{4}$ .

Tap for more steps...

Step 8.3.2.1.1

Apply the distributive property.

$y = 5 E i x \cdot x^{4} + C x^{4}$

Step 8.3.2.1.2

Multiply $x$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 8.3.2.1.2.1

Move $x^{4}$ .

$y = 5 E i (x^{4} x) + C x^{4}$

Step 8.3.2.1.2.2

Multiply $x^{4}$ by $x$ .

Tap for more steps...

Step 8.3.2.1.2.2.1

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

$y = 5 E i (x^{4} x^{1}) + C x^{4}$

Step 8.3.2.1.2.2.2

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

$y = 5 E i x^{4 + 1} + C x^{4}$

Step 8.3.2.1.2.3

Add $4$ and $1$ .

$y = 5 E i x^{5} + C x^{4}$

Step 8.3.2.1.3

Move $i$ .

$y = 5 E x^{5} i + C x^{4}$

Step 8.3.2.1.4

Move $E$ .

$y = 5 x^{5} E i + C x^{4}$