Calculus Examples

$\frac{d y}{d x} = y \sin (x)$ where $f (2 π) = 1$

Step 1

Separate the variables.

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Step 1.1

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

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y \sin (x))$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Factor $y$ out of $y \sin (x)$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y (\sin (x)))$

Step 1.2.2

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y \sin (x))$

Step 1.2.3

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = \sin (x)$

Step 1.3

Rewrite the equation.

$\frac{1}{y} d y = \sin (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int \sin (x) d x$

Step 2.2

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

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

Step 2.3

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

$\ln (| y |) + C_{1} = - \cos (x) + C_{2}$

Step 2.4

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

$\ln (| y |) = - \cos (x) + C$

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

$e^{- \cos (x) + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{- \cos (x) + C}$ .

$| y | = e^{- \cos (x) + C}$

Step 3.3.2

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

$y = \pm e^{- \cos (x) + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{- \cos (x) + C}$ as $e^{- \cos (x)} e^{C}$ .

$y = \pm e^{- \cos (x)} e^{C}$

Step 4.2

Reorder $e^{- \cos (x)}$ and $e^{C}$ .

$y = \pm e^{C} e^{- \cos (x)}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{- \cos (x)}$

Step 5

Use the initial condition to find the value of $C$ by substituting $2 π$ for $x$ and $1$ for $y$ in $y = C e^{- \cos (x)}$ .

$1 = C e^{- \cos (2 π)}$

Step 6

Solve for $C$ .

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Step 6.1

Rewrite the equation as $C e^{- \cos (2 π)} = 1$ .

$C e^{- \cos (2 π)} = 1$

Step 6.2

Simplify.

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Step 6.2.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

Step 6.2.2

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

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

Step 6.2.3

Multiply $- 1$ by $1$ .

$C e^{- 1} = 1$

Step 6.3

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

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Step 6.3.1

Divide each term in $C e^{- 1} = 1$ by $e^{- 1}$ .

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

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $e^{- 1}$ .

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Step 6.3.2.1.1

Cancel the common factor.

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

Step 6.3.2.1.2

Divide $C$ by $1$ .

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

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

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

$C = e$

Step 7

Substitute $e$ for $C$ in $y = C e^{- \cos (x)}$ and simplify.

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Step 7.1

Substitute $e$ for $C$ .

$y = e e^{- \cos (x)}$

Step 7.2

Multiply $e$ by $e^{- \cos (x)}$ .

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Step 7.2.1

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

$y = e^{1} e^{- \cos (x)}$

Step 7.2.2

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

$y = e^{1 - \cos (x)}$