Calculus Examples

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

Step 1

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

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Step 1.1

Reorder terms.

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

Step 1.2

Factor $y$ out of $- \frac{2 y}{x}$ .

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

Step 1.3

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

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

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 $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $- 1$ .

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

Step 2.2.4

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

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

Step 2.2.5

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 2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{2}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{2}}$ .

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Step 3.1

Multiply each term by $\frac{1}{x^{2}}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $- \frac{1}{x^{2}} \cdot \frac{2 y}{x}$ .

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Step 3.2.4.1

Multiply $\frac{2 y}{x}$ by $\frac{1}{x^{2}}$ .

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

Step 3.2.4.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.2.4.2.1

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

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Step 3.2.4.2.1.1

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

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

Step 3.2.4.2.1.2

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

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

Step 3.2.4.2.2

Add $1$ and $2$ .

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Simplify the expression.

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Step 7.1.1

Reorder $x^{2}$ and $1$ .

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

Step 7.1.2

Rewrite $1$ as $1^{2}$ .

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

Step 7.2

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C$ .

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

Step 8

Solve for $y$ .

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Step 8.1

Simplify the left side.

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Step 8.1.1

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

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

Step 8.2

Multiply both sides by $x^{2}$ .

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

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

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

Rewrite the expression.

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

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $(\arctan (x) + C) x^{2}$ .

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Step 8.3.2.1.1

Apply the distributive property.

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

Step 8.3.2.1.2

Simplify the expression.

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Step 8.3.2.1.2.1

Reorder factors in $\arctan (x) x^{2} + C x^{2}$ .

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

Step 8.3.2.1.2.2

Reorder $x^{2} \arctan (x)$ and $C x^{2}$ .

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