Calculus Examples

$2 (5) (0.5) + 2 (12) \frac{d y}{d t} = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d t}$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Multiply $2 (5) \cdot 0.5$ .

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Step 1.1.1.1.1

Multiply $2$ by $5$ .

$10 \cdot 0.5 + 2 (12) \frac{d y}{d t} = 0$

Step 1.1.1.1.2

Multiply $10$ by $0.5$ .

$5 + 2 (12) \frac{d y}{d t} = 0$

Step 1.1.1.2

Multiply $2$ by $12$ .

$5 + 24 \frac{d y}{d t} = 0$

Step 1.1.2

Subtract $5$ from both sides of the equation.

$24 \frac{d y}{d t} = - 5$

Step 1.1.3

Divide each term in $24 \frac{d y}{d t} = - 5$ by $24$ and simplify.

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Step 1.1.3.1

Divide each term in $24 \frac{d y}{d t} = - 5$ by $24$ .

$\frac{24 \frac{d y}{d t}}{24} = \frac{- 5}{24}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $24$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{24 \frac{d y}{d t}}{24} = \frac{- 5}{24}$

Step 1.1.3.2.1.2

Divide $\frac{d y}{d t}$ by $1$ .

$\frac{d y}{d t} = \frac{- 5}{24}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Cancel the common factor of $- 5$ and $24$ .

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Step 1.1.3.3.1.1

Rewrite $- 5$ as $1 (- 5)$ .

$\frac{d y}{d t} = \frac{1 (- 5)}{24}$

Step 1.1.3.3.1.2

Cancel the common factors.

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Step 1.1.3.3.1.2.1

Rewrite $24$ as $1 (24)$ .

$\frac{d y}{d t} = \frac{1 \cdot - 5}{1 \cdot 24}$

Step 1.1.3.3.1.2.2

Cancel the common factor.

$\frac{d y}{d t} = \frac{1 \cdot - 5}{1 \cdot 24}$

Step 1.1.3.3.1.2.3

Rewrite the expression.

$\frac{d y}{d t} = \frac{- 5}{24}$

Step 1.1.3.3.2

Move the negative in front of the fraction.

$\frac{d y}{d t} = - \frac{5}{24}$

Step 1.2

Rewrite the equation.

$d y = - \frac{5}{24} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - \frac{5}{24} d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - \frac{5}{24} d t$

Step 2.3

Apply the constant rule.

$y + C_{1} = - \frac{5}{24} t + C_{2}$

Step 2.4

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

$y = - \frac{5}{24} t + K$