Calculus Examples

$\frac{d y}{d t} = y \sin^{3} (t)$ , $y (0) = 1$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y}$ .

$\frac{1}{y} \frac{d y}{d t} = \frac{1}{y} (y \sin^{3} (t))$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Factor $y$ out of $y \sin^{3} (t)$ .

$\frac{1}{y} \frac{d y}{d t} = \frac{1}{y} (y (\sin^{3} (t)))$

Step 1.2.2

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d t} = \frac{1}{y} (y \sin^{3} (t))$

Step 1.2.3

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d t} = \sin^{3} (t)$

Step 1.3

Rewrite the equation.

$\frac{1}{y} d y = \sin^{3} (t) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int \sin^{3} (t) d t$

Step 2.2

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

$\ln (| y |) + C_{1} = \int \sin^{3} (t) d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Factor out $\sin^{2} (t)$ .

$\ln (| y |) + C_{1} = \int \sin^{2} (t) \sin (t) d t$

Step 2.3.2

Using the Pythagorean Identity, rewrite $\sin^{2} (t)$ as $1 - \cos^{2} (t)$ .

$\ln (| y |) + C_{1} = \int (1 - \cos^{2} (t)) \sin (t) d t$

Step 2.3.3

Let $u = \cos (t)$ . Then $d u = - \sin (t) d t$ , so $- \frac{1}{\sin (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.3.1

Let $u = \cos (t)$ . Find $\frac{d u}{d t}$ .

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Step 2.3.3.1.1

Differentiate $\cos (t)$ .

$\frac{d}{d t} [\cos (t)]$

Step 2.3.3.1.2

The derivative of $\cos (t)$ with respect to $t$ is $- \sin (t)$ .

$- \sin (t)$

Step 2.3.3.2

Rewrite the problem using $u$ and $d u$ .

$\ln (| y |) + C_{1} = \int - 1 + u^{2} d u$

Step 2.3.4

Split the single integral into multiple integrals.

$\ln (| y |) + C_{1} = \int - 1 d u + \int u^{2} d u$

Step 2.3.5

Apply the constant rule.

$\ln (| y |) + C_{1} = - u + C_{2} + \int u^{2} d u$

Step 2.3.6

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

$\ln (| y |) + C_{1} = - u + C_{2} + \frac{1}{3} u^{3} + C_{3}$

Step 2.3.7

Simplify.

$\ln (| y |) + C_{1} = - u + \frac{1}{3} u^{3} + C_{4}$

Step 2.3.8

Replace all occurrences of $u$ with $\cos (t)$ .

$\ln (| y |) + C_{1} = - \cos (t) + \frac{1}{3} \cos^{3} (t) + C_{4}$

Step 2.3.9

Reorder terms.

$\ln (| y |) + C_{1} = \frac{1}{3} \cos^{3} (t) - \cos (t) + C_{4}$

Step 2.4

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

$\ln (| y |) = \frac{1}{3} \cos^{3} (t) - \cos (t) + C$

Step 3

Solve for $y$ .

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Step 3.1

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

$e^{\ln (| y |)} = e^{\frac{1}{3} \cos^{3} (t) - \cos (t) + C}$

Step 3.2

Rewrite $\ln (| y |) = \frac{1}{3} \cos^{3} (t) - \cos (t) + C$ 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 $b^{y} = x$ .

$e^{\frac{1}{3} \cos^{3} (t) - \cos (t) + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{\frac{1}{3} \cdot \cos^{3} (t) - \cos (t) + C}$ .

$| y | = e^{\frac{1}{3} \cdot \cos^{3} (t) - \cos (t) + C}$

Step 3.3.2

Combine $\frac{1}{3}$ and $\cos^{3} (t)$ .

$| y | = e^{\frac{\cos^{3} (t)}{3} - \cos (t) + C}$

Step 3.3.3

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

$y = \pm e^{\frac{\cos^{3} (t)}{3} - \cos (t) + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{\frac{\cos^{3} (t)}{3} - \cos (t) + C}$ as $e^{\frac{\cos^{3} (t)}{3} - \cos (t)} e^{C}$ .

$y = \pm e^{\frac{\cos^{3} (t)}{3} - \cos (t)} e^{C}$

Step 4.2

Reorder $e^{\frac{\cos^{3} (t)}{3} - \cos (t)}$ and $e^{C}$ .

$y = \pm e^{C} e^{\frac{\cos^{3} (t)}{3} - \cos (t)}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{\frac{\cos^{3} (t)}{3} - \cos (t)}$

Step 5

Use the initial condition to find the value of $C$ by substituting $0$ for $t$ and $1$ for $y$ in $y = C e^{\frac{\cos^{3} (t)}{3} - \cos (t)}$ .

$1 = C e^{\frac{\cos^{3} (0)}{3} - \cos (0)}$

Step 6

Solve for $C$ .

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Step 6.1

Rewrite the equation as $C e^{\frac{\cos^{3} (0)}{3} - \cos (0)} = 1$ .

$C e^{\frac{\cos^{3} (0)}{3} - \cos (0)} = 1$

Step 6.2

Simplify.

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Step 6.2.1

The exact value of $\cos (0)$ is $1$ .

$C e^{\frac{\cos^{3} (0)}{3} - 1 \cdot 1} = 1$

Step 6.2.2

Multiply $- 1$ by $1$ .

$C e^{\frac{\cos^{3} (0)}{3} - 1} = 1$

Step 6.2.3

Simplify the numerator.

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Step 6.2.3.1

The exact value of $\cos (0)$ is $1$ .

$C e^{\frac{1^{3}}{3} - 1} = 1$

Step 6.2.3.2

One to any power is one.

$C e^{\frac{1}{3} - 1} = 1$

Step 6.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$C e^{\frac{1}{3} - 1 \cdot \frac{3}{3}} = 1$

Step 6.2.5

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

$C e^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} = 1$

Step 6.2.6

Combine the numerators over the common denominator.

$C e^{\frac{1 - 1 \cdot 3}{3}} = 1$

Step 6.2.7

Simplify the numerator.

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Step 6.2.7.1

Multiply $- 1$ by $3$ .

$C e^{\frac{1 - 3}{3}} = 1$

Step 6.2.7.2

Subtract $3$ from $1$ .

$C e^{\frac{- 2}{3}} = 1$

Step 6.3

Divide each term in $C e^{\frac{- 2}{3}} = 1$ by $e^{- \frac{2}{3}}$ and simplify.

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Step 6.3.1

Divide each term in $C e^{\frac{- 2}{3}} = 1$ by $e^{- \frac{2}{3}}$ .

$\frac{C e^{\frac{- 2}{3}}}{e^{- \frac{2}{3}}} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Factor $e^{- \frac{2}{3}}$ out of $C e^{\frac{- 2}{3}}$ .

$\frac{e^{- \frac{2}{3}} (C e^{\frac{0}{3}})}{e^{- \frac{2}{3}}} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

Multiply by $1$ .

$\frac{e^{- \frac{2}{3}} (C e^{\frac{0}{3}})}{e^{- \frac{2}{3}} \cdot 1} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.2.2

Cancel the common factor.

$\frac{e^{- \frac{2}{3}} (C e^{\frac{0}{3}})}{e^{- \frac{2}{3}} \cdot 1} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.2.3

Rewrite the expression.

$\frac{C e^{\frac{0}{3}}}{1} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.2.4

Divide $C e^{\frac{0}{3}}$ by $1$ .

$C e^{\frac{0}{3}} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.3

Cancel the common factor of $0$ and $3$ .

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Step 6.3.2.3.1

Factor $3$ out of $0$ .

$C e^{\frac{3 (0)}{3}} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.3.2

Cancel the common factors.

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Step 6.3.2.3.2.1

Factor $3$ out of $3$ .

$C e^{\frac{3 \cdot 0}{3 \cdot 1}} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.3.2.2

Cancel the common factor.

$C e^{\frac{3 \cdot 0}{3 \cdot 1}} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.3.2.3

Rewrite the expression.

$C e^{\frac{0}{1}} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.3.2.4

Divide $0$ by $1$ .

$C e^{0} = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.4

Simplify the expression.

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Step 6.3.2.4.1

Anything raised to $0$ is $1$ .

$C \cdot 1 = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.2.4.2

Multiply $C$ by $1$ .

$C = \frac{1}{e^{- \frac{2}{3}}}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Move $e^{- \frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$C = e^{\frac{2}{3}}$

Step 7

Substitute $e^{\frac{2}{3}}$ for $C$ in $y = C e^{\frac{\cos^{3} (t)}{3} - \cos (t)}$ and simplify.

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Step 7.1

Substitute $e^{\frac{2}{3}}$ for $C$ .

$y = e^{\frac{2}{3}} e^{\frac{\cos^{3} (t)}{3} - \cos (t)}$

Step 7.2

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

$y = e^{\frac{2}{3} + \frac{\cos^{3} (t)}{3} - \cos (t)}$