Calculus Examples

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

Step 1

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

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Step 1.1

Factor $y$ out of $\frac{3 y}{x}$ .

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

Step 1.2

Reorder $y$ and $\frac{3}{x}$ .

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $\frac{3}{x}$ .

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Step 2.2.1

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

$e^{3 \int \frac{1}{x} d x}$

Step 2.2.2

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

$e^{3 (\ln (| x |) + C)}$

Step 2.2.3

Simplify.

$e^{3 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{3 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{3})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{3}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$x^{3} y = e^{x} + C$

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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