Calculus Examples

$\frac{d y}{d x} + x y = 6 x \sqrt{y}$

Step 1

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

$\frac{d y}{d x} + x y = 6 x y^{\frac{1}{2}}$

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 $u_{1}$ as $v$ .

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

Step 5.2.2

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

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

Step 5.2.3

Replace all occurrences of $u_{1}$ 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 $\frac{d y}{d x} + x y = 6 x y^{\frac{1}{2}}$ .

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

Step 7

Solve the substituted differential equation.

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Step 7.1

Separate the variables.

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Step 7.1.1

Solve for $\frac{d v}{d x}$ .

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Step 7.1.1.1

Simplify $6 x {(v^{2})}^{\frac{1}{2}}$ .

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Step 7.1.1.1.1

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

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Step 7.1.1.1.1.1

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

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

Step 7.1.1.1.1.2

Cancel the common factor of $2$ .

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Step 7.1.1.1.1.2.1

Cancel the common factor.

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

Step 7.1.1.1.1.2.2

Rewrite the expression.

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

Step 7.1.1.1.2

Simplify.

$2 v \frac{d v}{d x} + x v^{2} = 6 x v$

Step 7.1.1.2

Subtract $x v^{2}$ from both sides of the equation.

$2 v \frac{d v}{d x} = 6 x v - x v^{2}$

Step 7.1.1.3

Divide each term in $2 v \frac{d v}{d x} = 6 x v - x v^{2}$ by $2 v$ and simplify.

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Step 7.1.1.3.1

Divide each term in $2 v \frac{d v}{d x} = 6 x v - x v^{2}$ by $2 v$ .

$\frac{2 v \frac{d v}{d x}}{2 v} = \frac{6 x v}{2 v} + \frac{- x v^{2}}{2 v}$

Step 7.1.1.3.2

Simplify the left side.

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Step 7.1.1.3.2.1

Cancel the common factor of $2$ .

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Step 7.1.1.3.2.1.1

Cancel the common factor.

$\frac{2 v \frac{d v}{d x}}{2 v} = \frac{6 x v}{2 v} + \frac{- x v^{2}}{2 v}$

Step 7.1.1.3.2.1.2

Rewrite the expression.

$\frac{v \frac{d v}{d x}}{v} = \frac{6 x v}{2 v} + \frac{- x v^{2}}{2 v}$

Step 7.1.1.3.2.2

Cancel the common factor of $v$ .

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Step 7.1.1.3.2.2.1

Cancel the common factor.

$\frac{v \frac{d v}{d x}}{v} = \frac{6 x v}{2 v} + \frac{- x v^{2}}{2 v}$

Step 7.1.1.3.2.2.2

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

$\frac{d v}{d x} = \frac{6 x v}{2 v} + \frac{- x v^{2}}{2 v}$

Step 7.1.1.3.3

Simplify the right side.

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Step 7.1.1.3.3.1

Simplify each term.

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Step 7.1.1.3.3.1.1

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

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Step 7.1.1.3.3.1.1.1

Factor $2$ out of $6 x v$ .

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

Step 7.1.1.3.3.1.1.2

Cancel the common factors.

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Step 7.1.1.3.3.1.1.2.1

Factor $2$ out of $2 v$ .

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

Step 7.1.1.3.3.1.1.2.2

Cancel the common factor.

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

Step 7.1.1.3.3.1.1.2.3

Rewrite the expression.

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

Step 7.1.1.3.3.1.2

Cancel the common factor of $v$ .

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Step 7.1.1.3.3.1.2.1

Cancel the common factor.

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

Step 7.1.1.3.3.1.2.2

Divide $3 x$ by $1$ .

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

Step 7.1.1.3.3.1.3

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

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Step 7.1.1.3.3.1.3.1

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

$\frac{d v}{d x} = 3 x + \frac{v (- x v)}{2 v}$

Step 7.1.1.3.3.1.3.2

Cancel the common factors.

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Step 7.1.1.3.3.1.3.2.1

Factor $v$ out of $2 v$ .

$\frac{d v}{d x} = 3 x + \frac{v (- x v)}{v \cdot 2}$

Step 7.1.1.3.3.1.3.2.2

Cancel the common factor.

$\frac{d v}{d x} = 3 x + \frac{v (- x v)}{v \cdot 2}$

Step 7.1.1.3.3.1.3.2.3

Rewrite the expression.

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

Step 7.1.1.3.3.1.4

Move the negative in front of the fraction.

$\frac{d v}{d x} = 3 x - \frac{x v}{2}$

Step 7.1.2

Factor.

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Step 7.1.2.1

Factor $x$ out of $3 x - \frac{x v}{2}$ .

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Step 7.1.2.1.1

Factor $x$ out of $3 x$ .

$\frac{d v}{d x} = x \cdot 3 - \frac{x v}{2}$

Step 7.1.2.1.2

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

$\frac{d v}{d x} = x \cdot 3 + x (- \frac{v}{2})$

Step 7.1.2.1.3

Factor $x$ out of $x \cdot 3 + x (- \frac{v}{2})$ .

$\frac{d v}{d x} = x (3 - \frac{v}{2})$

Step 7.1.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{d v}{d x} = x (3 \cdot \frac{2}{2} - \frac{v}{2})$

Step 7.1.2.3

Combine $3$ and $\frac{2}{2}$ .

$\frac{d v}{d x} = x (\frac{3 \cdot 2}{2} - \frac{v}{2})$

Step 7.1.2.4

Combine the numerators over the common denominator.

$\frac{d v}{d x} = x \frac{3 \cdot 2 - v}{2}$

Step 7.1.2.5

Multiply $3$ by $2$ .

$\frac{d v}{d x} = x \frac{6 - v}{2}$

Step 7.1.3

Multiply both sides by $\frac{2}{6 - v}$ .

$\frac{2}{6 - v} \frac{d v}{d x} = \frac{2}{6 - v} (x \frac{6 - v}{2})$

Step 7.1.4

Simplify.

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Step 7.1.4.1

Combine $x$ and $\frac{6 - v}{2}$ .

$\frac{2}{6 - v} \frac{d v}{d x} = \frac{2}{6 - v} \cdot \frac{x (6 - v)}{2}$

Step 7.1.4.2

Cancel the common factor of $2$ .

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Step 7.1.4.2.1

Cancel the common factor.

$\frac{2}{6 - v} \frac{d v}{d x} = \frac{2}{6 - v} \cdot \frac{x (6 - v)}{2}$

Step 7.1.4.2.2

Rewrite the expression.

$\frac{2}{6 - v} \frac{d v}{d x} = \frac{1}{6 - v} (x (6 - v))$

Step 7.1.4.3

Cancel the common factor of $6 - v$ .

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Step 7.1.4.3.1

Factor $6 - v$ out of $x (6 - v)$ .

$\frac{2}{6 - v} \frac{d v}{d x} = \frac{1}{6 - v} ((6 - v) x)$

Step 7.1.4.3.2

Cancel the common factor.

$\frac{2}{6 - v} \frac{d v}{d x} = \frac{1}{6 - v} ((6 - v) x)$

Step 7.1.4.3.3

Rewrite the expression.

$\frac{2}{6 - v} \frac{d v}{d x} = x$

Step 7.1.5

Rewrite the equation.

$\frac{2}{6 - v} d v = x d x$

Step 7.2

Integrate both sides.

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Step 7.2.1

Set up an integral on each side.

$\int \frac{2}{6 - v} d v = \int x d x$

Step 7.2.2

Integrate the left side.

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Step 7.2.2.1

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

$2 \int \frac{1}{6 - v} d v = \int x d x$

Step 7.2.2.2

Let $u_{2} = 6 - v$ . Then $d u_{2} = - d v$ , so $- d u_{2} = d v$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 7.2.2.2.1

Let $u_{2} = 6 - v$ . Find $\frac{d u_{2}}{d v}$ .

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Step 7.2.2.2.1.1

Rewrite.

$\frac{- 1}{1}$

Step 7.2.2.2.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 7.2.2.2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$2 \int \frac{- 1}{u_{2}} d u_{2} = \int x d x$

Step 7.2.2.3

Move the negative in front of the fraction.

$2 \int - \frac{1}{u_{2}} d u_{2} = \int x d x$

Step 7.2.2.4

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$2 (- \int \frac{1}{u_{2}} d u_{2}) = \int x d x$

Step 7.2.2.5

Multiply $- 1$ by $2$ .

$- 2 \int \frac{1}{u_{2}} d u_{2} = \int x d x$

Step 7.2.2.6

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$- 2 (\ln (| u_{2} |) + C_{1}) = \int x d x$

Step 7.2.2.7

Simplify.

$- 2 \ln (| u_{2} |) + C_{1} = \int x d x$

Step 7.2.2.8

Replace all occurrences of $u_{2}$ with $6 - v$ .

$- 2 \ln (| 6 - v |) + C_{1} = \int x d x$

Step 7.2.3

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

$- 2 \ln (| 6 - v |) + C_{1} = \frac{1}{2} x^{2} + C_{2}$

Step 7.2.4

Group the constant of integration on the right side as $C$ .

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

Step 7.3

Solve for $v$ .

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Step 7.3.1

Divide each term in $- 2 \ln (| 6 - v |) = \frac{1}{2} x^{2} + C$ by $- 2$ and simplify.

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Step 7.3.1.1

Divide each term in $- 2 \ln (| 6 - v |) = \frac{1}{2} x^{2} + C$ by $- 2$ .

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

Step 7.3.1.2

Simplify the left side.

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Step 7.3.1.2.1

Cancel the common factor of $- 2$ .

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Step 7.3.1.2.1.1

Cancel the common factor.

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

Step 7.3.1.2.1.2

Divide $\ln (| 6 - v |)$ by $1$ .

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

Step 7.3.1.3

Simplify the right side.

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Step 7.3.1.3.1

Simplify each term.

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Step 7.3.1.3.1.1

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

$\ln (| 6 - v |) = \frac{\frac{x^{2}}{2}}{- 2} + \frac{C}{- 2}$

Step 7.3.1.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$\ln (| 6 - v |) = \frac{x^{2}}{2} \cdot \frac{1}{- 2} + \frac{C}{- 2}$

Step 7.3.1.3.1.3

Combine.

$\ln (| 6 - v |) = \frac{x^{2} \cdot 1}{2 \cdot - 2} + \frac{C}{- 2}$

Step 7.3.1.3.1.4

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

$\ln (| 6 - v |) = \frac{x^{2}}{2 \cdot - 2} + \frac{C}{- 2}$

Step 7.3.1.3.1.5

Multiply $2$ by $- 2$ .

$\ln (| 6 - v |) = \frac{x^{2}}{- 4} + \frac{C}{- 2}$

Step 7.3.1.3.1.6

Move the negative in front of the fraction.

$\ln (| 6 - v |) = - \frac{x^{2}}{4} + \frac{C}{- 2}$

Step 7.3.1.3.1.7

Move the negative in front of the fraction.

$\ln (| 6 - v |) = - \frac{x^{2}}{4} - \frac{C}{2}$

Step 7.3.2

To solve for $v$ , rewrite the equation using properties of logarithms.

$e^{\ln (| 6 - v |)} = e^{- \frac{x^{2}}{4} - \frac{C}{2}}$

Step 7.3.3

Rewrite $\ln (| 6 - v |) = - \frac{x^{2}}{4} - \frac{C}{2}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- \frac{x^{2}}{4} - \frac{C}{2}} = | 6 - v |$

Step 7.3.4

Solve for $v$ .

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Step 7.3.4.1

Rewrite the equation as $| 6 - v | = e^{- \frac{x^{2}}{4} - \frac{C}{2}}$ .

$| 6 - v | = e^{- \frac{x^{2}}{4} - \frac{C}{2}}$

Step 7.3.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$6 - v = \pm e^{- \frac{x^{2}}{4} - \frac{C}{2}}$

Step 7.3.4.3

Subtract $6$ from both sides of the equation.

$- v = \pm e^{- \frac{x^{2}}{4} - \frac{C}{2}} - 6$

Step 7.3.4.4

Divide each term in $- v = \pm e^{- \frac{x^{2}}{4} - \frac{C}{2}} - 6$ by $- 1$ and simplify.

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Step 7.3.4.4.1

Divide each term in $- v = \pm e^{- \frac{x^{2}}{4} - \frac{C}{2}} - 6$ by $- 1$ .

$\frac{- v}{- 1} = \frac{\pm e^{- \frac{x^{2}}{4} - \frac{C}{2}}}{- 1} + \frac{- 6}{- 1}$

Step 7.3.4.4.2

Simplify the left side.

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Step 7.3.4.4.2.1

Dividing two negative values results in a positive value.

$\frac{v}{1} = \frac{\pm e^{- \frac{x^{2}}{4} - \frac{C}{2}}}{- 1} + \frac{- 6}{- 1}$

Step 7.3.4.4.2.2

Divide $v$ by $1$ .

$v = \frac{\pm e^{- \frac{x^{2}}{4} - \frac{C}{2}}}{- 1} + \frac{- 6}{- 1}$

Step 7.3.4.4.3

Simplify the right side.

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Step 7.3.4.4.3.1

Simplify each term.

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Step 7.3.4.4.3.1.1

Move the negative one from the denominator of $\frac{\pm e^{- \frac{x^{2}}{4} - \frac{C}{2}}}{- 1}$ .

$v = - 1 \cdot (\pm e^{- \frac{x^{2}}{4} - \frac{C}{2}}) + \frac{- 6}{- 1}$

Step 7.3.4.4.3.1.2

Rewrite $- 1 \cdot (\pm e^{- \frac{x^{2}}{4} - \frac{C}{2}})$ as $- (\pm e^{- \frac{x^{2}}{4} - \frac{C}{2}})$ .

$v = - (\pm e^{- \frac{x^{2}}{4} - \frac{C}{2}}) + \frac{- 6}{- 1}$

Step 7.3.4.4.3.1.3

Divide $- 6$ by $- 1$ .

$v = - (\pm e^{- \frac{x^{2}}{4} - \frac{C}{2}}) + 6$

Step 7.4

Group the constant terms together.

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Step 7.4.1

Simplify the constant of integration.

$v = - (\pm e^{- \frac{x^{2}}{4} + C}) + 6$

Step 7.4.2

Rewrite $e^{- \frac{x^{2}}{4} + C}$ as $e^{- \frac{x^{2}}{4}} e^{C}$ .

$v = - (\pm e^{- \frac{x^{2}}{4}} e^{C}) + 6$

Step 7.4.3

Reorder $e^{- \frac{x^{2}}{4}}$ and $e^{C}$ .

$v = - (\pm e^{C} e^{- \frac{x^{2}}{4}}) + 6$

Step 7.4.4

Combine constants with the plus or minus.

$v = - (C e^{- \frac{x^{2}}{4}}) + 6$

Step 8

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

$y^{\frac{1}{2}} = - (C e^{- \frac{x^{2}}{4}}) + 6$