Calculus Examples

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

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 $x y$ from both sides of the equation.

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

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

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

Step 1.4

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

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Step 1.4.1

Factor $x$ out of $- x y$ .

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

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

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

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 1.5

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

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Divide $3$ by $1$ .

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

Step 1.6

Factor $y$ out of $\frac{- 1 y}{x}$ .

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

Step 1.7

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

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

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 $\frac{1}{x}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

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 $\frac{1}{x}$ .

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

Step 3.2.5.2

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

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

Step 3.2.5.3

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

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

Step 3.2.5.4

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

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

Step 3.2.5.5

Add $1$ and $1$ .

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

Step 3.3

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 7.2

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

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

Step 7.3

Simplify.

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

Step 8

Solve for $y$ .

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Step 8.1

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

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

Step 8.2

Simplify $3 \ln (| x |)$ by moving $3$ inside the logarithm.

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

Step 8.3

Multiply both sides by $x$ .

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

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.

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

Step 8.4.1.1.2

Rewrite the expression.

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

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(\ln ({| x |}^{3}) + C) x$ .

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Step 8.4.2.1.1

Apply the distributive property.

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

Step 8.4.2.1.2

Simplify the expression.

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Step 8.4.2.1.2.1

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

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

Step 8.4.2.1.2.2

Reorder $x \ln ({| x |}^{3})$ and $C x$ .

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