Calculus Examples

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

Step 1

Rewrite the differential equation to fit the Bernoulli technique.

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Step 1.1

Subtract $\frac{2 y}{x}$ from both sides of the equation.

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $y^{2}$ .

$v = y^{- 1}$

Step 3

Solve the equation for $y$ .

$y = v^{- 1}$

Step 4

Take the derivative of $y$ with respect to $x$ .

$y' = v^{- 1}$

Step 5

Take the derivative of $v^{- 1}$ with respect to $x$ .

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Step 5.1

Take the derivative of $v^{- 1}$ .

$y' = \frac{d}{d x} [v^{- 1}]$

Step 5.2

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

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

Step 5.3

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1$ and $g (x) = v$ .

$y' = \frac{v \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [v]}{v^{2}}$

Step 5.4

Differentiate using the Constant Rule.

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Step 5.4.1

Multiply $- 1$ by $1$ .

$y' = \frac{v \frac{d}{d x} [1] - \frac{d}{d x} [v]}{v^{2}}$

Step 5.4.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$y' = \frac{v \cdot 0 - \frac{d}{d x} [v]}{v^{2}}$

Step 5.4.3

Simplify the expression.

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Step 5.4.3.1

Multiply $v$ by $0$ .

$y' = \frac{0 - \frac{d}{d x} [v]}{v^{2}}$

Step 5.4.3.2

Subtract $\frac{d}{d x} [v]$ from $0$ .

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

Step 5.4.3.3

Move the negative in front of the fraction.

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

Step 5.5

Rewrite $\frac{d}{d x} [v]$ as $v'$ .

$y' = - \frac{v'}{v^{2}}$

Step 6

Substitute $- \frac{v'}{v^{2}}$ for $\frac{d y}{d x}$ and $v^{- 1}$ for $y$ in the original equation $\frac{d y}{d x} - \frac{2}{x} y = - x^{2} y^{2}$ .

$- \frac{v'}{v^{2}} - \frac{2}{x} v^{- 1} = - x^{2} {(v^{- 1})}^{2}$

Step 7

Solve the substituted differential equation.

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Step 7.1

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

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Step 7.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{v^{2}} - \frac{2}{x} v^{- 1} = - x^{2} {(v^{- 1})}^{2}$ by $- v^{2}$ to eliminate the fractions.

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Step 7.1.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{v^{2}} - \frac{2}{x} v^{- 1} = - x^{2} {(v^{- 1})}^{2}$ by $- v^{2}$ .

$- \frac{\frac{d v}{d x}}{v^{2}} (- v^{2}) - \frac{2}{x} v^{- 1} (- v^{2}) = - x^{2} {(v^{- 1})}^{2} (- v^{2})$

Step 7.1.1.2

Simplify the left side.

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Step 7.1.1.2.1

Simplify each term.

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Step 7.1.1.2.1.1

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

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Step 7.1.1.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{v^{2}}$ into the numerator.

$\frac{- \frac{d v}{d x}}{v^{2}} (- v^{2}) - \frac{2}{x} v^{- 1} (- v^{2}) = - x^{2} {(v^{- 1})}^{2} (- v^{2})$

Step 7.1.1.2.1.1.2

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

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

Step 7.1.1.2.1.1.3

Cancel the common factor.

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

Step 7.1.1.2.1.1.4

Rewrite the expression.

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

Step 7.1.1.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} - \frac{2}{x} v^{- 1} (- v^{2}) = - x^{2} {(v^{- 1})}^{2} (- v^{2})$

Step 7.1.1.2.1.3

Multiply $\frac{d v}{d x}$ by $1$ .

$\frac{d v}{d x} - \frac{2}{x} v^{- 1} (- v^{2}) = - x^{2} {(v^{- 1})}^{2} (- v^{2})$

Step 7.1.1.2.1.4

Multiply $v^{- 1}$ by $v^{2}$ by adding the exponents.

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Step 7.1.1.2.1.4.1

Move $v^{2}$ .

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

Step 7.1.1.2.1.4.2

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

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

Step 7.1.1.2.1.4.3

Subtract $1$ from $2$ .

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

Step 7.1.1.2.1.5

Simplify $- \frac{2}{x} v^{1} \cdot - 1$ .

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

Step 7.1.1.2.1.6

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

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

Step 7.1.1.2.1.7

Move $2$ to the left of $v$ .

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

Step 7.1.1.2.1.8

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

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Step 7.1.1.2.1.8.1

Multiply $- 1$ by $- 1$ .

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

Step 7.1.1.2.1.8.2

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

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

Step 7.1.1.3

Simplify the right side.

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Step 7.1.1.3.1

Multiply the exponents in ${(v^{- 1})}^{2}$ .

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Step 7.1.1.3.1.1

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

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

Step 7.1.1.3.1.2

Multiply $- 1$ by $2$ .

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

Step 7.1.1.3.2

Multiply $v^{- 2}$ by $v^{2}$ by adding the exponents.

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Step 7.1.1.3.2.1

Move $v^{2}$ .

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

Step 7.1.1.3.2.2

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

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

Step 7.1.1.3.2.3

Subtract $2$ from $2$ .

$\frac{d v}{d x} + \frac{2 v}{x} = - x^{2} v^{0} \cdot - 1$

Step 7.1.1.3.3

Simplify $- x^{2} v^{0} \cdot - 1$ .

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

Step 7.1.1.3.4

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

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Step 7.1.1.3.4.1

Multiply $- 1$ by $- 1$ .

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

Step 7.1.1.3.4.2

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

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.2

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 7.2.1

Set up the integration.

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

Step 7.2.2

Integrate $\frac{2}{x}$ .

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Step 7.2.2.1

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

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

Step 7.2.2.2

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

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

Step 7.2.2.3

Simplify.

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

Step 7.2.3

Remove the constant of integration.

$e^{2 \ln (x)}$

Step 7.2.4

Use the logarithmic power rule.

$e^{\ln (x^{2})}$

Step 7.2.5

Exponentiation and log are inverse functions.

$x^{2}$

Step 7.3

Multiply each term by the integrating factor $x^{2}$ .

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Step 7.3.1

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

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

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Cancel the common factor of $x$ .

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Step 7.3.2.2.1

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

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

Step 7.3.2.2.2

Cancel the common factor.

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

Step 7.3.2.2.3

Rewrite the expression.

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

Step 7.3.2.3

Rewrite using the commutative property of multiplication.

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

Step 7.3.3

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

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Step 7.3.3.1

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

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

Step 7.3.3.2

Add $2$ and $2$ .

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

Step 7.4

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

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

Step 7.5

Set up an integral on each side.

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

Step 7.6

Integrate the left side.

$x^{2} v = \int x^{4} d x$

Step 7.7

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

$x^{2} v = \frac{1}{5} x^{5} + C$

Step 7.8

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

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Step 7.8.1

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

$\frac{x^{2} v}{x^{2}} = \frac{\frac{1}{5} x^{5}}{x^{2}} + \frac{C}{x^{2}}$

Step 7.8.2

Simplify the left side.

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Step 7.8.2.1

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

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Step 7.8.2.1.1

Cancel the common factor.

$\frac{x^{2} v}{x^{2}} = \frac{\frac{1}{5} x^{5}}{x^{2}} + \frac{C}{x^{2}}$

Step 7.8.2.1.2

Divide $v$ by $1$ .

$v = \frac{\frac{1}{5} x^{5}}{x^{2}} + \frac{C}{x^{2}}$

Step 7.8.3

Simplify the right side.

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Step 7.8.3.1

Simplify each term.

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Step 7.8.3.1.1

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

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Step 7.8.3.1.1.1

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

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

Step 7.8.3.1.1.2

Cancel the common factors.

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Step 7.8.3.1.1.2.1

Multiply by $1$ .

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

Step 7.8.3.1.1.2.2

Cancel the common factor.

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

Step 7.8.3.1.1.2.3

Rewrite the expression.

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

Step 7.8.3.1.1.2.4

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

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

Step 7.8.3.1.2

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

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

Step 8

Substitute $y^{- 1}$ for $v$ .

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