Calculus Examples

$2 (24) (0.6) + 2 (7) \frac{d y}{d t} = 0$

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 $2 (24) \cdot 0.6$ .

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Step 1.1.1.1.1

Multiply $2$ by $24$ .

$48 \cdot 0.6 + 2 (7) \frac{d y}{d t} = 0$

Step 1.1.1.1.2

Multiply $48$ by $0.6$ .

$28.8 + 2 (7) \frac{d y}{d t} = 0$

Step 1.1.1.2

Multiply $2$ by $7$ .

$28.8 + 14 \frac{d y}{d t} = 0$

Step 1.1.2

Subtract $28.8$ from both sides of the equation.

$14 \frac{d y}{d t} = - 28.8$

Step 1.1.3

Divide each term in $14 \frac{d y}{d t} = - 28.8$ by $14$ and simplify.

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Step 1.1.3.1

Divide each term in $14 \frac{d y}{d t} = - 28.8$ by $14$ .

$\frac{14 \frac{d y}{d t}}{14} = \frac{- 28.8}{14}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $14$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{14 \frac{d y}{d t}}{14} = \frac{- 28.8}{14}$

Step 1.1.3.2.1.2

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

$\frac{d y}{d t} = \frac{- 28.8}{14}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Divide $- 28.8$ by $14$ .

$\frac{d y}{d t} = - 2.0\overline{571428}$

Step 1.2

Rewrite the equation.

$d y = - 2.0\overline{571428} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - 2.0\overline{571428} d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - 2.0\overline{571428} d t$

Step 2.3

Apply the constant rule.

$y + C_{1} = - 2.0\overline{571428} t + C_{2}$

Step 2.4

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

$y = - 2.0\overline{571428} t + K$