Calculus Examples

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

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

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

Step 1.2

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

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

Step 2

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 2.1

Set up the integration.

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Simplify.

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

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

Step 2.5

Exponentiation and log are inverse functions.

$x^{2}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 4

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

$\frac{d}{d x} [x^{2} y] = 1$

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

$x^{2} y = \int d x$

Step 7

Apply the constant rule.

$x^{2} y = x + C$

Step 8

Divide each term in $x^{2} y = x + C$ by $x^{2}$ and simplify.

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Step 8.1

Divide each term in $x^{2} y = x + C$ by $x^{2}$ .

$\frac{x^{2} y}{x^{2}} = \frac{x}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

$\frac{x^{2} y}{x^{2}} = \frac{x}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

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

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Step 8.3.1.1

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

$y = \frac{x^{1}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3.1.2

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

$y = \frac{x \cdot 1}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3.1.3

Cancel the common factors.

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Step 8.3.1.3.1

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

$y = \frac{x \cdot 1}{x \cdot x} + \frac{C}{x^{2}}$

Step 8.3.1.3.2

Cancel the common factor.

$y = \frac{x \cdot 1}{x \cdot x} + \frac{C}{x^{2}}$

Step 8.3.1.3.3

Rewrite the expression.

$y = \frac{1}{x} + \frac{C}{x^{2}}$