Calculus Examples

$2 w t \frac{d t}{d w} = t^{2} - 2 w^{3}$

Step 1

Let $v = t^{2}$ . Substitute $v$ for all occurrences of $t^{2}$ .

$2 w t \frac{d t}{d w} = v - 2 w^{3}$

Step 2

Find $\frac{d v}{d w}$ by differentiating $t^{2}$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = w^{2}$ and $g (w) = t$ .

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Step 2.1.1

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

$\frac{d v}{d w} = \frac{d}{d u} [u^{2}] \frac{d}{d w} [t]$

Step 2.1.2

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

$\frac{d v}{d w} = 2 u \frac{d}{d w} [t]$

Step 2.1.3

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

$\frac{d v}{d w} = 2 t \frac{d}{d w} [t]$

Step 2.2

Rewrite $\frac{d}{d w} [t]$ as $t'$ .

$\frac{d v}{d w} = 2 t t'$

Step 3

Substitute the derivative back in to the differential equation.

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Step 3.1

Factor $2 t$ out of $2 w t$ .

$2 t (w) \frac{\frac{d v}{d w}}{2 t} = v - 2 w^{3}$

Step 3.2

Cancel the common factor.

$2 t w \frac{\frac{d v}{d w}}{2 t} = v - 2 w^{3}$

Step 3.3

Rewrite the expression.

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

Step 4

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

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Step 4.1

Subtract $v$ from both sides of the equation.

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

Step 4.2

Divide each term in $w \frac{d v}{d w} - v = - 2 w^{3}$ by $w$ .

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

Step 4.3

Cancel the common factor of $w$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

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

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

Step 4.4

Cancel the common factor of $w^{3}$ and $w$ .

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Step 4.4.1

Factor $w$ out of $- 2 w^{3}$ .

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

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

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

Step 4.4.2.2

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

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

Step 4.4.2.3

Cancel the common factor.

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

Step 4.4.2.4

Rewrite the expression.

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

Step 4.4.2.5

Divide $- 2 w^{2}$ by $1$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 5

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

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Step 5.1

Set up the integration.

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

Step 5.2

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

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Step 5.2.1

Split the fraction into multiple fractions.

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Simplify.

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

Step 5.3

Remove the constant of integration.

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

Step 5.4

Use the logarithmic power rule.

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

Step 5.5

Exponentiation and log are inverse functions.

$w^{- 1}$

Step 5.6

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

$\frac{1}{w}$

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify each term.

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Step 6.2.1

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

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

Step 6.2.2

Move the negative in front of the fraction.

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

Step 6.2.3

Rewrite using the commutative property of multiplication.

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

Step 6.2.4

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

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

Step 6.2.5

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

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Step 6.2.5.1

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

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

Step 6.2.5.2

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

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

Step 6.2.5.3

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

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

Step 6.2.5.4

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

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

Step 6.2.5.5

Add $1$ and $1$ .

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

Step 6.3

Rewrite using the commutative property of multiplication.

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

Step 6.4

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

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

Step 6.5

Cancel the common factor of $w$ .

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Step 6.5.1

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

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

Step 6.5.2

Cancel the common factor.

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

Step 6.5.3

Rewrite the expression.

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

Step 7

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

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

Step 8

Set up an integral on each side.

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

Step 9

Integrate the left side.

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

Step 10

Integrate the right side.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

Simplify the answer.

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Step 10.3.1

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

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

Step 10.3.2

Simplify.

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Step 10.3.2.1

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

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

Step 10.3.2.2

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

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Step 10.3.2.2.1

Factor $2$ out of $- 2$ .

$\frac{1}{w} v = \frac{2 \cdot - 1}{2} w^{2} + C$

Step 10.3.2.2.2

Cancel the common factors.

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Step 10.3.2.2.2.1

Factor $2$ out of $2$ .

$\frac{1}{w} v = \frac{2 \cdot - 1}{2 (1)} w^{2} + C$

Step 10.3.2.2.2.2

Cancel the common factor.

$\frac{1}{w} v = \frac{2 \cdot - 1}{2 \cdot 1} w^{2} + C$

Step 10.3.2.2.2.3

Rewrite the expression.

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

Step 10.3.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 11

Solve for $v$ .

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Step 11.1

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

$\frac{v}{w} = - w^{2} + C$

Step 11.2

Multiply both sides by $w$ .

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

Step 11.3

Simplify.

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Step 11.3.1

Simplify the left side.

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Step 11.3.1.1

Cancel the common factor of $w$ .

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Step 11.3.1.1.1

Cancel the common factor.

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

Step 11.3.1.1.2

Rewrite the expression.

$v = (- w^{2} + C) w$

Step 11.3.2

Simplify the right side.

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Step 11.3.2.1

Simplify $(- w^{2} + C) w$ .

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Step 11.3.2.1.1

Apply the distributive property.

$v = - w^{2} w + C w$

Step 11.3.2.1.2

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

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Step 11.3.2.1.2.1

Move $w$ .

$v = - (w \cdot w^{2}) + C w$

Step 11.3.2.1.2.2

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

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Step 11.3.2.1.2.2.1

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

$v = - (w^{1} w^{2}) + C w$

Step 11.3.2.1.2.2.2

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

$v = - w^{1 + 2} + C w$

Step 11.3.2.1.2.3

Add $1$ and $2$ .

$v = - w^{3} + C w$

Step 12

Replace all occurrences of $v$ with $t^{2}$ .

$t^{2} = - w^{3} + C w$

Step 13

Solve for $t$ .

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Step 13.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{- w^{3} + C w}$

Step 13.2

Factor $w$ out of $- w^{3} + C w$ .

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Step 13.2.1

Factor $w$ out of $- w^{3}$ .

$t = \pm \sqrt{w (- w^{2}) + C w}$

Step 13.2.2

Factor $w$ out of $C w$ .

$t = \pm \sqrt{w (- w^{2}) + w C}$

Step 13.2.3

Factor $w$ out of $w (- w^{2}) + w C$ .

$t = \pm \sqrt{w (- w^{2} + C)}$

Step 13.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 13.3.1

First, use the positive value of the $\pm$ to find the first solution.

$t = \sqrt{w (- w^{2} + C)}$

Step 13.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - \sqrt{w (- w^{2} + C)}$

Step 13.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = \sqrt{w (- w^{2} + C)}$

$t = - \sqrt{w (- w^{2} + C)}$

$t = \sqrt{w (- w^{2} + C)}$

$t = - \sqrt{w (- w^{2} + C)}$

$t = \sqrt{w (- w^{2} + C)}$

$t = - \sqrt{w (- w^{2} + C)}$