Calculus Examples

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

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

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

Step 1.2

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

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

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

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

Step 1.4.2.2

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

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

Step 1.4.2.3

Cancel the common factor.

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

Step 1.4.2.4

Rewrite the expression.

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

Step 1.4.2.5

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

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

Step 1.5

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

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Step 1.5.1

Factor $x$ out of $3 x^{2}$ .

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

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

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

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

Step 1.5.2.2

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

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

Step 1.5.2.3

Cancel the common factor.

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

Step 1.5.2.4

Rewrite the expression.

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

Step 1.5.2.5

Divide $3 x$ by $1$ .

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

Step 1.6

Cancel the common factor of $x$ .

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Step 1.6.1

Cancel the common factor.

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

Step 1.6.2

Divide $- 2$ by $1$ .

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

Step 1.7

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

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

Step 1.8

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

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

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

Integrate $\frac{- 1}{x}$ .

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Step 2.2.1

Split the fraction into multiple fractions.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Simplify.

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 1})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 1}$

Step 2.6

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

$\frac{1}{x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

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

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

Step 3.2.5.3

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

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

Step 3.2.5.4

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

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

Step 3.2.5.5

Add $1$ and $1$ .

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

Step 3.3

Simplify each term.

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Step 3.3.1

Cancel the common factor of $x$ .

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Step 3.3.1.1

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

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

Step 3.3.1.2

Cancel the common factor.

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

Step 3.3.1.3

Rewrite the expression.

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

Step 3.3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3.3

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

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

Step 3.3.4

Cancel the common factor of $x$ .

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Step 3.3.4.1

Cancel the common factor.

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

Step 3.3.4.2

Rewrite the expression.

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

Step 3.3.5

Combine $\frac{1}{x}$ and $- 2$ .

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

Step 3.3.6

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x} y] d x = \int x + 3 - \frac{2}{x} d x$

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

Apply the constant rule.

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

Multiply $2$ by $- 1$ .

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

Step 7.7

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

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

Step 7.8

Simplify.

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

Step 8

Solve for $y$ .

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Step 8.1

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

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

Step 8.2

Simplify each term.

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

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

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

Step 8.3

Multiply both sides by $x$ .

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

Step 8.4

Simplify.

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

Cancel the common factor of $x$ .

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Step 8.4.1.1.1

Cancel the common factor.

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

Step 8.4.1.1.2

Rewrite the expression.

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

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(\frac{x^{2}}{2} + 3 x - \ln (x^{2}) + C) x$ .

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Step 8.4.2.1.1

Apply the distributive property.

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

Step 8.4.2.1.2

Simplify.

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Step 8.4.2.1.2.1

Multiply $\frac{x^{2}}{2} x$ .

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Step 8.4.2.1.2.1.1

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

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

Step 8.4.2.1.2.1.2

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

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Step 8.4.2.1.2.1.2.1

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

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Step 8.4.2.1.2.1.2.1.1

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

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

Step 8.4.2.1.2.1.2.1.2

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

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

Step 8.4.2.1.2.1.2.2

Add $2$ and $1$ .

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

Step 8.4.2.1.2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 8.4.2.1.2.2.1

Move $x$ .

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

Step 8.4.2.1.2.2.2

Multiply $x$ by $x$ .

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

Step 8.4.2.1.3

Reorder factors in $\frac{x^{3}}{2} + 3 x^{2} - \ln (x^{2}) x + C x$ .

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

Step 8.4.2.1.4

Move $- x \ln (x^{2})$ .

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