Calculus Examples

$3 \frac{d y}{d θ} + \sin (θ) = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d θ}$ .

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Step 1.1.1

Subtract $\sin (θ)$ from both sides of the equation.

$3 \frac{d y}{d θ} = - \sin (θ)$

Step 1.1.2

Divide each term in $3 \frac{d y}{d θ} = - \sin (θ)$ by $3$ and simplify.

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Step 1.1.2.1

Divide each term in $3 \frac{d y}{d θ} = - \sin (θ)$ by $3$ .

$\frac{3 \frac{d y}{d θ}}{3} = \frac{- \sin (θ)}{3}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $3$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{3 \frac{d y}{d θ}}{3} = \frac{- \sin (θ)}{3}$

Step 1.1.2.2.1.2

Divide $\frac{d y}{d θ}$ by $1$ .

$\frac{d y}{d θ} = \frac{- \sin (θ)}{3}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Move the negative in front of the fraction.

$\frac{d y}{d θ} = - \frac{\sin (θ)}{3}$

Step 1.2

Rewrite the equation.

$d y = - \frac{\sin (θ)}{3} d θ$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - \frac{\sin (θ)}{3} d θ$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - \frac{\sin (θ)}{3} d θ$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$y + C_{1} = - \int \frac{\sin (θ)}{3} d θ$

Step 2.3.2

Since $\frac{1}{3}$ is constant with respect to $θ$ , move $\frac{1}{3}$ out of the integral.

$y + C_{1} = - (\frac{1}{3} \int \sin (θ) d θ)$

Step 2.3.3

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

$y + C_{1} = - \frac{1}{3} (- \cos (θ) + C_{2})$

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Simplify.

$y + C_{1} = - \frac{1}{3} (- \cos (θ)) + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Multiply $- 1$ by $- 1$ .

$y + C_{1} = 1 (\frac{1}{3}) \cos (θ) + C_{2}$

Step 2.3.4.2.2

Multiply $\frac{1}{3}$ by $1$ .

$y + C_{1} = \frac{1}{3} \cos (θ) + C_{2}$

Step 2.3.4.2.3

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

$y + C_{1} = \frac{\cos (θ)}{3} + C_{2}$

Step 2.3.4.3

Reorder terms.

$y + C_{1} = \frac{1}{3} \cos (θ) + C_{2}$

Step 2.4

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

$y = \frac{1}{3} \cos (θ) + K$