Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{2}{x}$ .

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Step 1.1

Set up the integration.

$e^{\int \frac{2}{x} d x}$

Step 1.2

Integrate $\frac{2}{x}$ .

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Step 1.2.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$e^{2 \int \frac{1}{x} d x}$

Step 1.2.2

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

$e^{2 (\ln (| x |) + C)}$

Step 1.2.3

Simplify.

$e^{2 \ln (| x |) + C}$

Step 1.3

Remove the constant of integration.

$e^{2 \ln (x)}$

Step 1.4

Use the logarithmic power rule.

$e^{\ln (x^{2})}$

Step 1.5

Exponentiation and log are inverse functions.

$x^{2}$

Step 2

Multiply each term by the integrating factor $x^{2}$ .

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Step 2.1

Multiply each term by $x^{2}$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{2}{x}$ and $y$ .

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

Step 2.2.2

Cancel the common factor of $x$ .

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Step 2.2.2.1

Factor $x$ out of $x^{2}$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.2.3

Rewrite using the commutative property of multiplication.

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

Step 2.3

Cancel the common factor of $x^{2}$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

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

Step 3

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

$\frac{d}{d x} [x^{2} y] = \sin (3 x)$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} y] d x = \int \sin (3 x) d x$

Step 5

Integrate the left side.

$x^{2} y = \int \sin (3 x) d x$

Step 6

Integrate the right side.

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Step 6.1

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 6.1.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 6.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 6.1.1.4

Multiply $3$ by $1$ .

$3$

Step 6.1.2

Rewrite the problem using $u$ and $d u$ .

$x^{2} y = \int \sin (u) \frac{1}{3} d u$

Step 6.2

Combine $\sin (u)$ and $\frac{1}{3}$ .

$x^{2} y = \int \frac{\sin (u)}{3} d u$

Step 6.3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$x^{2} y = \frac{1}{3} \int \sin (u) d u$

Step 6.4

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$x^{2} y = \frac{1}{3} (- \cos (u) + C)$

Step 6.5

Simplify.

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Step 6.5.1

Simplify.

$x^{2} y = \frac{1}{3} (- \cos (u)) + C$

Step 6.5.2

Combine $\frac{1}{3}$ and $\cos (u)$ .

$x^{2} y = - \frac{\cos (u)}{3} + C$

Step 6.6

Replace all occurrences of $u$ with $3 x$ .

$x^{2} y = - \frac{\cos (3 x)}{3} + C$

Step 6.7

Reorder terms.

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

Step 7

Divide each term in $x^{2} y = - \frac{1}{3} \cos (3 x) + C$ by $x^{2}$ and simplify.

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Step 7.1

Divide each term in $x^{2} y = - \frac{1}{3} \cos (3 x) + C$ by $x^{2}$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $x^{2}$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

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 $\cos (3 x)$ and $\frac{1}{3}$ .

$y = \frac{- \frac{\cos (3 x)}{3}}{x^{2}} + \frac{C}{x^{2}}$

Step 7.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{\cos (3 x)}{3} \cdot \frac{1}{x^{2}} + \frac{C}{x^{2}}$

Step 7.3.1.3

Multiply $\frac{1}{x^{2}}$ by $\frac{\cos (3 x)}{3}$ .

$y = - \frac{\cos (3 x)}{x^{2} \cdot 3} + \frac{C}{x^{2}}$

Step 7.3.1.4

Move $3$ to the left of $x^{2}$ .

$y = - \frac{\cos (3 x)}{3 x^{2}} + \frac{C}{x^{2}}$