Calculus Examples

$\frac{d y}{d t} + \csc (y) = 0$

Step 1

Separate the variables.

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Step 1.1

Subtract $\csc (y)$ from both sides of the equation.

$\frac{d y}{d t} = - \csc (y)$

Step 1.2

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

$\frac{1}{\csc (y)} \frac{d y}{d t} = \frac{1}{\csc (y)} (- \csc (y))$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\csc (y)} \frac{d y}{d t} = - \frac{1}{\csc (y)} \csc (y)$

Step 1.3.2

Cancel the common factor of $\csc (y)$ .

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Step 1.3.2.1

Move the leading negative in $- \frac{1}{\csc (y)}$ into the numerator.

$\frac{1}{\csc (y)} \frac{d y}{d t} = \frac{- 1}{\csc (y)} \csc (y)$

Step 1.3.2.2

Cancel the common factor.

$\frac{1}{\csc (y)} \frac{d y}{d t} = \frac{- 1}{\csc (y)} \csc (y)$

Step 1.3.2.3

Rewrite the expression.

$\frac{1}{\csc (y)} \frac{d y}{d t} = - 1$

Step 1.4

Rewrite the equation.

$\frac{1}{\csc (y)} d y = - d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\csc (y)} d y = \int - 1 d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Rewrite $\csc (y)$ in terms of sines and cosines.

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

Step 2.2.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (y)}$ .

$\int 1 \sin (y) d y = \int - 1 d t$

Step 2.2.1.3

Multiply $\sin (y)$ by $1$ .

$\int \sin (y) d y = \int - 1 d t$

Step 2.2.2

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

$- \cos (y) + C_{1} = \int - 1 d t$

Step 2.3

Apply the constant rule.

$- \cos (y) + C_{1} = - t + C_{2}$

Step 2.4

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

$- \cos (y) = - t + K$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $- \cos (y) = - t + K$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- \cos (y) = - t + K$ by $- 1$ .

$\frac{- \cos (y)}{- 1} = \frac{- t}{- 1} + \frac{K}{- 1}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (y)}{1} = \frac{- t}{- 1} + \frac{K}{- 1}$

Step 3.1.2.2

Divide $\cos (y)$ by $1$ .

$\cos (y) = \frac{- t}{- 1} + \frac{K}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Dividing two negative values results in a positive value.

$\cos (y) = \frac{t}{1} + \frac{K}{- 1}$

Step 3.1.3.1.2

Divide $t$ by $1$ .

$\cos (y) = t + \frac{K}{- 1}$

Step 3.1.3.1.3

Move the negative one from the denominator of $\frac{K}{- 1}$ .

$\cos (y) = t - 1 \cdot K$

Step 3.1.3.1.4

Rewrite $- 1 \cdot K$ as $- K$ .

$\cos (y) = t - K$

Step 3.2

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$y = \arccos (t - K)$

Step 4

Simplify the constant of integration.

$y = \arccos (t + K)$