Calculus Examples

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

Step 1

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

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Step 1.1

Divide each term in $x^{2} \frac{d y}{d x} + x y = 2$ by $x^{2}$ .

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.3

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

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Step 1.3.1

Factor $x$ out of $x y$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

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

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

Step 1.5

Reorder $y$ and $\frac{1}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{x}$ .

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Step 2.1

Set up the integration.

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

Step 2.2

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

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

Step 2.3

Remove the constant of integration.

$e^{\ln (x)}$

Step 2.4

Exponentiation and log are inverse functions.

$x$

Step 3

Multiply each term by the integrating factor $x$ .

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Step 3.1

Multiply each term by $x$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

$x y = 2 \int \frac{1}{x} d x$

Step 7.2

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

$x y = 2 (\ln (| x |) + C)$

Step 7.3

Simplify.

$x y = 2 \ln (| x |) + C$

Step 8

Divide each term in $x y = 2 \ln (| x |) + C$ by $x$ and simplify.

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Step 8.1

Divide each term in $x y = 2 \ln (| x |) + C$ by $x$ .

$\frac{x y}{x} = \frac{2 \ln (| x |)}{x} + \frac{C}{x}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $x$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{2 \ln (| x |)}{x} + \frac{C}{x}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 \ln (| x |)}{x} + \frac{C}{x}$

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

Simplify $2 \ln (| x |)$ by moving $2$ inside the logarithm.

$y = \frac{\ln ({| x |}^{2})}{x} + \frac{C}{x}$

Step 8.3.1.2

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

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