Calculus Examples

$\frac{d y}{d x} + \frac{y}{x} = \sqrt{y}$

Step 1

Rewrite the differential equation to fit the Bernoulli technique.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 1.2

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

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

Step 1.3

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

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

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

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

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 $2 v \frac{d v}{d x} + \frac{1}{x} v^{2} = {(v^{2})}^{\frac{1}{2}}$ by $2 v$ and simplify.

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Step 7.1.1.1

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

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

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.

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

Step 7.1.1.2.1.1.2

Rewrite the expression.

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

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.

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

Step 7.1.1.2.1.2.2

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

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

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

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

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 $2 v$ .

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

Step 7.1.1.2.1.3.2.2

Cancel the common factor.

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

Step 7.1.1.2.1.3.2.3

Rewrite the expression.

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

Step 7.1.1.2.1.4

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

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

Step 7.1.1.2.1.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.1.1.2.1.6

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

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

Step 7.1.1.2.1.7

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

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

Step 7.1.1.3

Simplify the right side.

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Step 7.1.1.3.1

Simplify the numerator.

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Step 7.1.1.3.1.1

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

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Step 7.1.1.3.1.1.1

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

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

Step 7.1.1.3.1.1.2

Cancel the common factor of $2$ .

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Step 7.1.1.3.1.1.2.1

Cancel the common factor.

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

Step 7.1.1.3.1.1.2.2

Rewrite the expression.

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

Step 7.1.1.3.1.2

Simplify.

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

Step 7.1.1.3.2

Cancel the common factor of $v$ .

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Step 7.1.1.3.2.1

Cancel the common factor.

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

Step 7.1.1.3.2.2

Rewrite the expression.

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

Step 7.1.2

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

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

Step 7.1.3

Reorder $v$ and $\frac{1}{2 x}$ .

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

Step 7.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{2 x}$ .

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Step 7.2.1

Set up the integration.

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

Step 7.2.2

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

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Step 7.2.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 7.2.2.2

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

$e^{\frac{1}{2} (\ln (| x |) + C)}$

Step 7.2.2.3

Simplify.

$e^{\frac{1}{2} \ln (| x |) + C}$

Step 7.2.3

Remove the constant of integration.

$e^{\frac{1}{2} \ln (x)}$

Step 7.2.4

Use the logarithmic power rule.

$e^{\ln (x^{\frac{1}{2}})}$

Step 7.2.5

Exponentiation and log are inverse functions.

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

Step 7.3

Multiply each term by the integrating factor $x^{\frac{1}{2}}$ .

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Step 7.3.1

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

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

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

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

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

Step 7.3.2.2

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

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

Step 7.3.2.3

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 7.3.2.4

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.3.2.4.1

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

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

Step 7.3.2.4.2

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

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Step 7.3.2.4.2.1

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

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

Step 7.3.2.4.2.2

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

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

Step 7.3.2.4.3

Write $1$ as a fraction with a common denominator.

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

Step 7.3.2.4.4

Combine the numerators over the common denominator.

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

Step 7.3.2.4.5

Add $- 1$ and $2$ .

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

Step 7.3.3

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

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

Step 7.4

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

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

Step 7.5

Set up an integral on each side.

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

Step 7.6

Integrate the left side.

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

Step 7.7

Integrate the right side.

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Step 7.7.1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 7.7.2

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

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

Step 7.7.3

Simplify the answer.

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Step 7.7.3.1

Rewrite $\frac{1}{2} (\frac{2}{3} x^{\frac{3}{2}} + C)$ as $\frac{1}{2} \cdot \frac{2}{3} x^{\frac{3}{2}} + C$ .

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

Step 7.7.3.2

Simplify.

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Step 7.7.3.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

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

Step 7.7.3.2.2

Multiply $2$ by $3$ .

$x^{\frac{1}{2}} v = \frac{2}{6} x^{\frac{3}{2}} + C$

Step 7.7.3.2.3

Cancel the common factor of $2$ and $6$ .

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Step 7.7.3.2.3.1

Factor $2$ out of $2$ .

$x^{\frac{1}{2}} v = \frac{2 (1)}{6} x^{\frac{3}{2}} + C$

Step 7.7.3.2.3.2

Cancel the common factors.

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Step 7.7.3.2.3.2.1

Factor $2$ out of $6$ .

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

Step 7.7.3.2.3.2.2

Cancel the common factor.

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

Step 7.7.3.2.3.2.3

Rewrite the expression.

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

Step 7.8

Divide each term in $x^{\frac{1}{2}} v = \frac{1}{3} x^{\frac{3}{2}} + C$ by $x^{\frac{1}{2}}$ and simplify.

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Step 7.8.1

Divide each term in $x^{\frac{1}{2}} v = \frac{1}{3} x^{\frac{3}{2}} + C$ by $x^{\frac{1}{2}}$ .

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

Step 7.8.2

Simplify the left side.

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Step 7.8.2.1

Cancel the common factor.

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

Step 7.8.2.2

Divide $v$ by $1$ .

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

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

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 7.8.3.1.2

Multiply $x^{\frac{3}{2}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.8.3.1.2.1

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

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

Step 7.8.3.1.2.2

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

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

Step 7.8.3.1.2.3

Combine the numerators over the common denominator.

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

Step 7.8.3.1.2.4

Add $- 1$ and $3$ .

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

Step 7.8.3.1.2.5

Divide $2$ by $2$ .

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

Step 7.8.3.1.3

Simplify $\frac{1}{3} x^{1}$ .

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

Step 7.8.3.1.4

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

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

Step 8

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

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