Calculus Examples

$\frac{d y}{d x} - \frac{x^{2} - 2 y}{x} = 0$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Split the fraction $\frac{x^{2} - 2 y}{x}$ into two fractions.

$\frac{d y}{d x} - (\frac{x^{2}}{x} + \frac{- 2 y}{x}) = 0$

Step 1.1.2

Apply the distributive property.

$\frac{d y}{d x} - \frac{x^{2}}{x} - \frac{- 2 y}{x} = 0$

Step 1.1.3

Add $\frac{x^{2}}{x}$ to both sides of the equation.

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

Step 1.2

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

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

Step 1.3

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

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

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

Simplify.

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Step 2.2.1.1

Move the negative in front of the fraction.

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

Step 2.2.1.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.1.3

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

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

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

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

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

Step 2.2.4

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 $x^{2}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

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

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

Step 3.2.4

Cancel the common factor of $x$ .

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Step 3.2.4.1

Move the leading negative in $- \frac{y \cdot 2}{x}$ into the numerator.

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

Step 3.2.4.2

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

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

Step 3.2.4.3

Cancel the common factor.

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

Step 3.2.4.4

Rewrite the expression.

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

Step 3.2.5

Multiply $2$ by $- 1$ .

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

Step 3.2.6

Multiply $- 2$ by $- 1$ .

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

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

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Step 3.4.1

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

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Step 3.4.1.1

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

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

Step 3.4.1.2

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

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

Step 3.4.2

Add $1$ and $2$ .

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

Step 4

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

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

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} y] d x = \int x^{3} d x$

Step 6

Integrate the left side.

$x^{2} y = \int x^{3} d x$

Step 7

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$x^{2} y = \frac{1}{4} x^{4} + C$

Step 8

Divide each term in $x^{2} y = \frac{1}{4} x^{4} + C$ by $x^{2}$ and simplify.

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Step 8.1

Divide each term in $x^{2} y = \frac{1}{4} x^{4} + C$ by $x^{2}$ .

$\frac{x^{2} y}{x^{2}} = \frac{\frac{1}{4} x^{4}}{x^{2}} + \frac{C}{x^{2}}$

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.

$\frac{x^{2} y}{x^{2}} = \frac{\frac{1}{4} x^{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{4} x^{4}}{x^{2}} + \frac{C}{x^{2}}$

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 $x^{4}$ and $x^{2}$ .

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Step 8.3.1.1.1

Factor $x^{2}$ out of $\frac{1}{4} x^{4}$ .

$y = \frac{x^{2} (\frac{1}{4} x^{2})}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

$y = \frac{x^{2} (\frac{1}{4} x^{2})}{x^{2} \cdot 1} + \frac{C}{x^{2}}$

Step 8.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{2} (\frac{1}{4} x^{2})}{x^{2} \cdot 1} + \frac{C}{x^{2}}$

Step 8.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{4} x^{2}}{1} + \frac{C}{x^{2}}$

Step 8.3.1.1.2.4

Divide $\frac{1}{4} x^{2}$ by $1$ .

$y = \frac{1}{4} x^{2} + \frac{C}{x^{2}}$

Step 8.3.1.2

Combine $\frac{1}{4}$ and $x^{2}$ .

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