Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 3$ .

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Step 1.1

Set up the integration.

$e^{\int 3 d x}$

Step 1.2

Apply the constant rule.

$e^{3 x + C}$

Step 1.3

Remove the constant of integration.

$e^{3 x}$

Step 2

Multiply each term by the integrating factor $e^{3 x}$ .

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Step 2.1

Multiply each term by $e^{3 x}$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Rewrite using the commutative property of multiplication.

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

Step 2.4

Multiply $e^{3 x}$ by $e^{- 3 x}$ by adding the exponents.

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Step 2.4.1

Move $e^{- 3 x}$ .

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

Step 2.4.2

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

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

Step 2.4.3

Add $- 3 x$ and $3 x$ .

$e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 3 e^{0} x^{3}$

Step 2.5

Simplify $3 e^{0} x^{3}$ .

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

Step 2.6

Reorder factors in $e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 3 x^{3}$ .

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

$e^{3 x} y = \int 3 x^{3} d x$

Step 6

Integrate the right side.

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Step 6.1

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

$e^{3 x} y = 3 \int x^{3} d x$

Step 6.2

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

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

Step 6.3

Simplify the answer.

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Step 6.3.1

Rewrite $3 (\frac{1}{4} x^{4} + C)$ as $3 (\frac{1}{4}) x^{4} + C$ .

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

Step 6.3.2

Combine $3$ and $\frac{1}{4}$ .

$e^{3 x} y = \frac{3}{4} x^{4} + C$

Step 7

Divide each term in $e^{3 x} y = \frac{3}{4} x^{4} + C$ by $e^{3 x}$ and simplify.

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Step 7.1

Divide each term in $e^{3 x} y = \frac{3}{4} x^{4} + C$ by $e^{3 x}$ .

$\frac{e^{3 x} y}{e^{3 x}} = \frac{\frac{3}{4} x^{4}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{3 x} y}{e^{3 x}} = \frac{\frac{3}{4} x^{4}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{3}{4} x^{4}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Combine $\frac{3}{4}$ and $x^{4}$ .

$y = \frac{\frac{3 x^{4}}{4}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{3 x^{4}}{4} \cdot \frac{1}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.3.1.3

Combine.

$y = \frac{3 x^{4} \cdot 1}{4 e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.3.1.4

Multiply $3$ by $1$ .

$y = \frac{3 x^{4}}{4 e^{3 x}} + \frac{C}{e^{3 x}}$