Calculus Examples

$\frac{1}{3} \cdot \frac{d y}{d x} = y + \frac{1}{3} x^{2} e^{3 x}$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$\frac{1}{3} \frac{d y}{d x} - y = \frac{1}{3} x^{2} e^{3 x}$

Step 1.2

Multiply each term in $\frac{1}{3} \frac{d y}{d x} - y = \frac{1}{3} x^{2} e^{3 x}$ by $3$ .

$\frac{1}{3} \frac{d y}{d x} \cdot 3 - y \cdot 3 = \frac{1}{3} x^{2} e^{3 x} \cdot 3$

Step 1.3

Combine $\frac{1}{3}$ and $\frac{d y}{d x}$ .

$\frac{\frac{d y}{d x}}{3} \cdot 3 - y \cdot 3 = \frac{1}{3} x^{2} e^{3 x} \cdot 3$

Step 1.4

Cancel the common factor of $3$ .

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Step 1.4.1

Cancel the common factor.

$\frac{\frac{d y}{d x}}{3} \cdot 3 - y \cdot 3 = \frac{1}{3} x^{2} e^{3 x} \cdot 3$

Step 1.4.2

Rewrite the expression.

$\frac{d y}{d x} - y \cdot 3 = \frac{1}{3} x^{2} e^{3 x} \cdot 3$

Step 1.5

Multiply $3$ by $- 1$ .

$\frac{d y}{d x} - 3 y = \frac{1}{3} x^{2} e^{3 x} \cdot 3$

Step 1.6

Combine $\frac{1}{3}$ and $x^{2}$ .

$\frac{d y}{d x} - 3 y = \frac{x^{2}}{3} e^{3 x} \cdot 3$

Step 1.7

Combine $\frac{x^{2}}{3}$ and $e^{3 x}$ .

$\frac{d y}{d x} - 3 y = \frac{x^{2} e^{3 x}}{3} \cdot 3$

Step 1.8

Combine $\frac{x^{2} e^{3 x}}{3}$ and $3$ .

$\frac{d y}{d x} - 3 y = \frac{3}{3} x^{2} e^{3 x}$

Step 1.9

Cancel the common factor of $3$ .

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Step 1.9.1

Cancel the common factor.

$\frac{d y}{d x} - 3 y = \frac{3}{3} x^{2} e^{3 x}$

Step 1.9.2

Rewrite the expression.

$\frac{d y}{d x} - 3 y = 1 x^{2} e^{3 x}$

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int - 3 d x}$

Step 2.2

Apply the constant rule.

$e^{- 3 x + C}$

Step 2.3

Remove the constant of integration.

$e^{- 3 x}$

Step 3

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

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Step 3.1

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

$e^{- 3 x} \frac{d y}{d x} + e^{- 3 x} (- 3 y) = e^{- 3 x} (1 x^{2} e^{3 x})$

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{- 3 x} \frac{d y}{d x} - 3 e^{- 3 x} y = e^{- 3 x} (1 x^{2} e^{3 x})$

Step 3.3

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

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Step 3.3.1

Move $e^{3 x}$ .

$e^{- 3 x} \frac{d y}{d x} - 3 e^{- 3 x} y = e^{3 x} e^{- 3 x} (1 x^{2})$

Step 3.3.2

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

$e^{- 3 x} \frac{d y}{d x} - 3 e^{- 3 x} y = e^{3 x - 3 x} (1 x^{2})$

Step 3.3.3

Subtract $3 x$ from $3 x$ .

$e^{- 3 x} \frac{d y}{d x} - 3 e^{- 3 x} y = e^{0} (1 x^{2})$

Step 3.4

Simplify $e^{0} (1 x^{2})$ .

$e^{- 3 x} \frac{d y}{d x} - 3 e^{- 3 x} y = x^{2}$

Step 3.5

Reorder factors in $e^{- 3 x} \frac{d y}{d x} - 3 e^{- 3 x} y = x^{2}$ .

$e^{- 3 x} \frac{d y}{d x} - 3 y e^{- 3 x} = x^{2}$

Step 4

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

$\frac{d}{d x} [e^{- 3 x} y] = x^{2}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- 3 x} y] d x = \int x^{2} d x$

Step 6

Integrate the left side.

$e^{- 3 x} y = \int x^{2} d x$

Step 7

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 8

Divide each term in $e^{- 3 x} y = \frac{1}{3} x^{3} + C$ by $e^{- 3 x}$ and simplify.

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Step 8.1

Divide each term in $e^{- 3 x} y = \frac{1}{3} x^{3} + C$ by $e^{- 3 x}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 8.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.3.1.3

Combine.

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

Step 8.3.1.4

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

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