Calculus Examples

$x^{2} (\frac{d y}{d x}) - 2 x y = x^{4} + 3$

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 $x^{2} \frac{d y}{d x} - 2 x y = x^{4} + 3$ by $x^{2}$ .

$\frac{x^{2} \frac{d y}{d x}}{x^{2}} + \frac{- 2 x y}{x^{2}} = \frac{x^{4}}{x^{2}} + \frac{3}{x^{2}}$

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

$\frac{x^{2} \frac{d y}{d x}}{x^{2}} + \frac{- 2 x y}{x^{2}} = \frac{x^{4}}{x^{2}} + \frac{3}{x^{2}}$

Step 1.2.2

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

$\frac{d y}{d x} + \frac{- 2 x y}{x^{2}} = \frac{x^{4}}{x^{2}} + \frac{3}{x^{2}}$

Step 1.3

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

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Step 1.3.1

Factor $x$ out of $- 2 x y$ .

$\frac{d y}{d x} + \frac{x (- 2 y)}{x^{2}} = \frac{x^{4}}{x^{2}} + \frac{3}{x^{2}}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

$\frac{d y}{d x} + \frac{x (- 2 y)}{x \cdot x} = \frac{x^{4}}{x^{2}} + \frac{3}{x^{2}}$

Step 1.3.2.2

Cancel the common factor.

$\frac{d y}{d x} + \frac{x (- 2 y)}{x \cdot x} = \frac{x^{4}}{x^{2}} + \frac{3}{x^{2}}$

Step 1.3.2.3

Rewrite the expression.

$\frac{d y}{d x} + \frac{- 2 y}{x} = \frac{x^{4}}{x^{2}} + \frac{3}{x^{2}}$

Step 1.4

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 1.4.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Multiply by $1$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 1.4.2.4

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

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

Step 1.5

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

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

Step 1.6

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

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

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

Split the fraction into multiple fractions.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $2$ by $- 1$ .

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

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

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Step 3.2.5.2.1

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

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Step 3.2.5.2.1.1

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

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

Step 3.2.5.2.1.2

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

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

Step 3.2.5.2.2

Add $1$ and $2$ .

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

Step 3.3

Simplify each term.

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Step 3.3.1

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

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.3.2

Combine.

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

Step 3.3.3

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

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Step 3.3.3.1

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

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

Step 3.3.3.2

Add $2$ and $2$ .

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{2 y}{x^{3}} = 1 + \frac{1 \cdot 3}{x^{4}}$

Step 3.3.4

Multiply $3$ by $1$ .

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{2 y}{x^{3}} = 1 + \frac{3}{x^{4}}$

Step 4

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

$\frac{d}{d x} [\frac{1}{x^{2}} y] = 1 + \frac{3}{x^{4}}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{2}} y] d x = \int 1 + \frac{3}{x^{4}} d x$

Step 6

Integrate the left side.

$\frac{1}{x^{2}} y = \int 1 + \frac{3}{x^{4}} d x$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$\frac{1}{x^{2}} y = \int d x + \int \frac{3}{x^{4}} d x$

Step 7.2

Apply the constant rule.

$\frac{1}{x^{2}} y = x + C + \int \frac{3}{x^{4}} d x$

Step 7.3

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

$\frac{1}{x^{2}} y = x + C + 3 \int \frac{1}{x^{4}} d x$

Step 7.4

Apply basic rules of exponents.

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Step 7.4.1

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{x^{2}} y = x + C + 3 \int {(x^{4})}^{- 1} d x$

Step 7.4.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 7.4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{x^{2}} y = x + C + 3 \int x^{4 \cdot - 1} d x$

Step 7.4.2.2

Multiply $4$ by $- 1$ .

$\frac{1}{x^{2}} y = x + C + 3 \int x^{- 4} d x$

Step 7.5

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

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

Step 7.6

Simplify.

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Step 7.6.1

Simplify.

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Step 7.6.1.1

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

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

Step 7.6.1.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 7.6.2

Simplify.

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

Step 7.6.3

Move the negative in front of the fraction.

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

Step 8

Solve for $y$ .

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Step 8.1

Move all terms containing variables to the left side of the equation.

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Step 8.1.1

Subtract $x$ from both sides of the equation.

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

Step 8.1.2

Add $\frac{1}{x^{3}}$ to both sides of the equation.

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

Step 8.1.3

Subtract $C$ from both sides of the equation.

$\frac{1}{x^{2}} y - x + \frac{1}{x^{3}} - C = 0$

Step 8.1.4

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

$\frac{y}{x^{2}} - x + \frac{1}{x^{3}} - C = 0$

Step 8.2

Move all terms not containing $y$ to the right side of the equation.

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Step 8.2.1

Add $x$ to both sides of the equation.

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

Step 8.2.2

Subtract $\frac{1}{x^{3}}$ from both sides of the equation.

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

Step 8.2.3

Add $C$ to both sides of the equation.

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

Step 8.3

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

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

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 $x^{2}$ .

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Step 8.4.1.1.1

Cancel the common factor.

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

Step 8.4.1.1.2

Rewrite the expression.

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

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(x - \frac{1}{x^{3}} + C) x^{2}$ .

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Step 8.4.2.1.1

Apply the distributive property.

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

Step 8.4.2.1.2

Simplify.

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Step 8.4.2.1.2.1

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

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Step 8.4.2.1.2.1.1

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

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Step 8.4.2.1.2.1.1.1

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

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

Step 8.4.2.1.2.1.1.2

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

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

Step 8.4.2.1.2.1.2

Add $1$ and $2$ .

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

Step 8.4.2.1.2.2

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

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Step 8.4.2.1.2.2.1

Move the leading negative in $- \frac{1}{x^{3}}$ into the numerator.

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

Step 8.4.2.1.2.2.2

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

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

Step 8.4.2.1.2.2.3

Cancel the common factor.

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

Step 8.4.2.1.2.2.4

Rewrite the expression.

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

Step 8.4.2.1.3

Move the negative in front of the fraction.

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

Step 8.4.2.1.4

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

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