Calculus Examples

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

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

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Cancel the common factor.

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

Step 1.3.2.4

Rewrite the expression.

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

Step 1.3.2.5

Divide $x^{2}$ by $1$ .

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

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Divide $- 1$ by $1$ .

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

Step 1.5

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

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

Step 1.6

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

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

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 $2$ 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 $x^{2}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.3

Simplify each term.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

Add $2$ and $2$ .

$x^{2} \frac{d y}{d x} + 2 x y = x^{4} + x^{2} \cdot - 1$

Step 3.3.2

Move $- 1$ to the left of $x^{2}$ .

$x^{2} \frac{d y}{d x} + 2 x y = x^{4} - 1 \cdot x^{2}$

Step 3.3.3

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

$x^{2} \frac{d y}{d x} + 2 x y = x^{4} - x^{2}$

Step 4

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

$\frac{d}{d x} [x^{2} y] = x^{4} - x^{2}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} y] d x = \int x^{4} - x^{2} d x$

Step 6

Integrate the left side.

$x^{2} y = \int x^{4} - x^{2} d x$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$x^{2} y = \int x^{4} d x + \int - x^{2} d x$

Step 7.2

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$x^{2} y = \frac{1}{5} x^{5} + C + \int - x^{2} d x$

Step 7.3

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

$x^{2} y = \frac{1}{5} x^{5} + C - \int x^{2} d x$

Step 7.4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$x^{2} y = \frac{1}{5} x^{5} + C - (\frac{1}{3} x^{3} + C)$

Step 7.5

Simplify.

$x^{2} y = \frac{1}{5} x^{5} - \frac{1}{3} x^{3} + C$

Step 8

Divide each term in $x^{2} y = \frac{1}{5} x^{5} - \frac{1}{3} x^{3} + C$ by $x^{2}$ and simplify.

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Step 8.1

Divide each term in $x^{2} y = \frac{1}{5} x^{5} - \frac{1}{3} x^{3} + C$ by $x^{2}$ .

$\frac{x^{2} y}{x^{2}} = \frac{\frac{1}{5} x^{5}}{x^{2}} + \frac{- \frac{1}{3} x^{3}}{x^{2}} + \frac{C}{x^{2}}$

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.

$\frac{x^{2} y}{x^{2}} = \frac{\frac{1}{5} x^{5}}{x^{2}} + \frac{- \frac{1}{3} x^{3}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{5} x^{5}}{x^{2}} + \frac{- \frac{1}{3} x^{3}}{x^{2}} + \frac{C}{x^{2}}$

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

Cancel the common factor of $x^{5}$ and $x^{2}$ .

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Step 8.3.1.1.1

Factor $x^{2}$ out of $\frac{1}{5} x^{5}$ .

$y = \frac{x^{2} (\frac{1}{5} x^{3})}{x^{2}} + \frac{- \frac{1}{3} x^{3}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

$y = \frac{x^{2} (\frac{1}{5} x^{3})}{x^{2} \cdot 1} + \frac{- \frac{1}{3} x^{3}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{2} (\frac{1}{5} x^{3})}{x^{2} \cdot 1} + \frac{- \frac{1}{3} x^{3}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{5} x^{3}}{1} + \frac{- \frac{1}{3} x^{3}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3.1.1.2.4

Divide $\frac{1}{5} x^{3}$ by $1$ .

$y = \frac{1}{5} x^{3} + \frac{- \frac{1}{3} x^{3}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3.1.2

Combine $\frac{1}{5}$ and $x^{3}$ .

$y = \frac{x^{3}}{5} + \frac{- \frac{1}{3} x^{3}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3.1.3

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 8.3.1.3.1

Factor $x^{2}$ out of $- \frac{1}{3} x^{3}$ .

$y = \frac{x^{3}}{5} + \frac{x^{2} (- \frac{1}{3} x)}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3.1.3.2

Cancel the common factors.

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Step 8.3.1.3.2.1

Multiply by $1$ .

$y = \frac{x^{3}}{5} + \frac{x^{2} (- \frac{1}{3} x)}{x^{2} \cdot 1} + \frac{C}{x^{2}}$

Step 8.3.1.3.2.2

Cancel the common factor.

$y = \frac{x^{3}}{5} + \frac{x^{2} (- \frac{1}{3} x)}{x^{2} \cdot 1} + \frac{C}{x^{2}}$

Step 8.3.1.3.2.3

Rewrite the expression.

$y = \frac{x^{3}}{5} + \frac{- \frac{1}{3} x}{1} + \frac{C}{x^{2}}$

Step 8.3.1.3.2.4

Divide $- \frac{1}{3} x$ by $1$ .

$y = \frac{x^{3}}{5} - \frac{1}{3} x + \frac{C}{x^{2}}$

Step 8.3.1.4

Combine $x$ and $\frac{1}{3}$ .

$y = \frac{x^{3}}{5} - \frac{x}{3} + \frac{C}{x^{2}}$