Calculus Examples

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

Step 1

Rewrite the differential equation to fit the Bernoulli technique.

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Step 1.1

Divide each term in $x \frac{d y}{d x} - 2 y = 4 x^{3} y^{\frac{1}{2}}$ by $x$ .

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

Factor $x$ out of $4 x^{3} y^{\frac{1}{2}}$ .

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

Step 1.4

Cancel the common factors.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Cancel the common factor.

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

Step 1.4.4

Rewrite the expression.

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

Step 1.4.5

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 2

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

$v = y^{\frac{1}{2}}$

Step 3

Solve the equation for $y$ .

$y = v^{2}$

Step 4

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

$y' = v^{2}$

Step 5

Take the derivative of $v^{2}$ with respect to $x$ .

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Step 5.1

Take the derivative of $v^{2}$ .

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

Step 5.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = v$ .

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Step 5.2.1

To apply the Chain Rule, set $u$ as $v$ .

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

Step 5.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 5.2.3

Replace all occurrences of $u$ with $v$ .

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

Step 5.3

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

$y' = 2 v v'$

Step 6

Substitute $2 v v'$ for $\frac{d y}{d x}$ and $v^{2}$ for $y$ in the original equation $\frac{d y}{d x} + \frac{- 2}{x} y = 4 x^{2} y^{\frac{1}{2}}$ .

$2 v v' + \frac{- 2}{x} v^{2} = 4 x^{2} {(v^{2})}^{\frac{1}{2}}$

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

Divide each term in $2 v \frac{d v}{d x} + \frac{- 2}{x} v^{2} = 4 x^{2} {(v^{2})}^{\frac{1}{2}}$ by $2 v$ and simplify.

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Step 7.1.1.1

Divide each term in $2 v \frac{d v}{d x} + \frac{- 2}{x} v^{2} = 4 x^{2} {(v^{2})}^{\frac{1}{2}}$ by $2 v$ .

$\frac{2 v \frac{d v}{d x}}{2 v} + \frac{\frac{- 2}{x} v^{2}}{2 v} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

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 $2$ .

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Step 7.1.1.2.1.1.1

Cancel the common factor.

$\frac{2 v \frac{d v}{d x}}{2 v} + \frac{\frac{- 2}{x} v^{2}}{2 v} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.1.2

Rewrite the expression.

$\frac{v \frac{d v}{d x}}{v} + \frac{\frac{- 2}{x} v^{2}}{2 v} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.2

Cancel the common factor of $v$ .

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Step 7.1.1.2.1.2.1

Cancel the common factor.

$\frac{v \frac{d v}{d x}}{v} + \frac{\frac{- 2}{x} v^{2}}{2 v} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.2.2

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

$\frac{d v}{d x} + \frac{\frac{- 2}{x} v^{2}}{2 v} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.3

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

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Step 7.1.1.2.1.3.1

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

$\frac{d v}{d x} + \frac{v (\frac{- 2}{x} v)}{2 v} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.3.2

Cancel the common factors.

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Step 7.1.1.2.1.3.2.1

Factor $v$ out of $2 v$ .

$\frac{d v}{d x} + \frac{v (\frac{- 2}{x} v)}{v \cdot 2} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.3.2.2

Cancel the common factor.

$\frac{d v}{d x} + \frac{v (\frac{- 2}{x} v)}{v \cdot 2} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.3.2.3

Rewrite the expression.

$\frac{d v}{d x} + \frac{\frac{- 2}{x} v}{2} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.4

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

$\frac{d v}{d x} + \frac{\frac{- 2 v}{x}}{2} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.5

Move the negative in front of the fraction.

$\frac{d v}{d x} + \frac{- \frac{2 v}{x}}{2} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{d v}{d x} - \frac{2 v}{x} \cdot \frac{1}{2} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.7

Cancel the common factor of $2$ .

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Step 7.1.1.2.1.7.1

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

$\frac{d v}{d x} + \frac{- 2 v}{x} \cdot \frac{1}{2} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.7.2

Factor $2$ out of $- 2 v$ .

$\frac{d v}{d x} + \frac{2 (- v)}{x} \cdot \frac{1}{2} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.7.3

Cancel the common factor.

$\frac{d v}{d x} + \frac{2 (- v)}{x} \cdot \frac{1}{2} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.7.4

Rewrite the expression.

$\frac{d v}{d x} + \frac{- v}{x} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.2.1.8

Move the negative in front of the fraction.

$\frac{d v}{d x} - \frac{v}{x} = \frac{4 x^{2} {(v^{2})}^{\frac{1}{2}}}{2 v}$

Step 7.1.1.3

Simplify the right side.

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Step 7.1.1.3.1

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

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

Step 7.1.1.3.2

Cancel the common factors.

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Step 7.1.1.3.2.1

Factor $2$ out of $2 v$ .

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

Step 7.1.1.3.2.2

Cancel the common factor.

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

Step 7.1.1.3.2.3

Rewrite the expression.

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

Step 7.1.1.3.3

Simplify the numerator.

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Step 7.1.1.3.3.1

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

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Step 7.1.1.3.3.1.1

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

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

Step 7.1.1.3.3.1.2

Cancel the common factor of $2$ .

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Step 7.1.1.3.3.1.2.1

Cancel the common factor.

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

Step 7.1.1.3.3.1.2.2

Rewrite the expression.

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

Step 7.1.1.3.3.2

Simplify.

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

Step 7.1.1.3.4

Cancel the common factor of $v$ .

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Step 7.1.1.3.4.1

Cancel the common factor.

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

Step 7.1.1.3.4.2

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

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.2

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 7.2.1

Set up the integration.

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

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

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

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

Step 7.2.2.3

Simplify.

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

Step 7.2.3

Remove the constant of integration.

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

Step 7.2.4

Use the logarithmic power rule.

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

Step 7.2.5

Exponentiation and log are inverse functions.

$x^{- 1}$

Step 7.2.6

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

$\frac{1}{x}$

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Rewrite using the commutative property of multiplication.

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

Step 7.3.2.3

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

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

Step 7.3.2.4

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

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Step 7.3.2.4.1

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

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

Step 7.3.2.4.2

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

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

Step 7.3.2.4.3

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

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

Step 7.3.2.4.4

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

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

Step 7.3.2.4.5

Add $1$ and $1$ .

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

Step 7.3.3

Rewrite using the commutative property of multiplication.

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

Step 7.3.4

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

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

Step 7.3.5

Cancel the common factor of $x$ .

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Step 7.3.5.1

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

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

Step 7.3.5.2

Cancel the common factor.

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

Step 7.3.5.3

Rewrite the expression.

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

Step 7.4

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

$\frac{d}{d x} [\frac{1}{x} v] = 2 x$

Step 7.5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x} v] d x = \int 2 x d x$

Step 7.6

Integrate the left side.

$\frac{1}{x} v = \int 2 x d x$

Step 7.7

Integrate the right side.

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Step 7.7.1

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

$\frac{1}{x} v = 2 \int x d x$

Step 7.7.2

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

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

Step 7.7.3

Simplify the answer.

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Step 7.7.3.1

Rewrite $2 (\frac{1}{2} x^{2} + C)$ as $2 (\frac{1}{2}) x^{2} + C$ .

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

Step 7.7.3.2

Simplify.

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Step 7.7.3.2.1

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

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

Step 7.7.3.2.2

Cancel the common factor of $2$ .

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Step 7.7.3.2.2.1

Cancel the common factor.

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

Step 7.7.3.2.2.2

Rewrite the expression.

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

Step 7.7.3.2.3

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

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

Step 7.8

Solve for $v$ .

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Step 7.8.1

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

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

Step 7.8.2

Multiply both sides by $x$ .

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

Step 7.8.3

Simplify.

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Step 7.8.3.1

Simplify the left side.

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Step 7.8.3.1.1

Cancel the common factor of $x$ .

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Step 7.8.3.1.1.1

Cancel the common factor.

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

Step 7.8.3.1.1.2

Rewrite the expression.

$v = (x^{2} + C) x$

Step 7.8.3.2

Simplify the right side.

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Step 7.8.3.2.1

Simplify $(x^{2} + C) x$ .

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Step 7.8.3.2.1.1

Apply the distributive property.

$v = x^{2} x + C x$

Step 7.8.3.2.1.2

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

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Step 7.8.3.2.1.2.1

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

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Step 7.8.3.2.1.2.1.1

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

$v = x^{2} x^{1} + C x$

Step 7.8.3.2.1.2.1.2

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

$v = x^{2 + 1} + C x$

Step 7.8.3.2.1.2.2

Add $2$ and $1$ .

$v = x^{3} + C x$

Step 8

Substitute $y^{\frac{1}{2}}$ for $v$ .

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