Calculus Examples

$4 (5) \frac{d y}{d t} + 4 (- 1) (- 8) - 3 \cdot - 8 + 15 {(- 1)}^{2} \frac{d y}{d t} = 0$

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 $4 (5) \frac{d y}{d t} + 4 (- 1) \cdot - 8 - 3 \cdot - 8 + 15 {(- 1)}^{2} \frac{d y}{d t}$ .

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Step 1.1.1.1

Simplify each term.

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Step 1.1.1.1.1

Multiply $4$ by $5$ .

$20 \frac{d y}{d t} + 4 (- 1) \cdot - 8 - 3 \cdot - 8 + 15 {(- 1)}^{2} \frac{d y}{d t} = 0$

Step 1.1.1.1.2

Multiply $4 (- 1) \cdot - 8$ .

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Step 1.1.1.1.2.1

Multiply $4$ by $- 1$ .

$20 \frac{d y}{d t} - 4 \cdot - 8 - 3 \cdot - 8 + 15 {(- 1)}^{2} \frac{d y}{d t} = 0$

Step 1.1.1.1.2.2

Multiply $- 4$ by $- 8$ .

$20 \frac{d y}{d t} + 32 - 3 \cdot - 8 + 15 {(- 1)}^{2} \frac{d y}{d t} = 0$

Step 1.1.1.1.3

Multiply $- 3$ by $- 8$ .

$20 \frac{d y}{d t} + 32 + 24 + 15 {(- 1)}^{2} \frac{d y}{d t} = 0$

Step 1.1.1.1.4

Raise $- 1$ to the power of $2$ .

$20 \frac{d y}{d t} + 32 + 24 + 15 \cdot 1 \frac{d y}{d t} = 0$

Step 1.1.1.1.5

Multiply $15$ by $1$ .

$20 \frac{d y}{d t} + 32 + 24 + 15 \frac{d y}{d t} = 0$

Step 1.1.1.2

Simplify by adding terms.

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Step 1.1.1.2.1

Add $20 \frac{d y}{d t}$ and $15 \frac{d y}{d t}$ .

$35 \frac{d y}{d t} + 32 + 24 = 0$

Step 1.1.1.2.2

Add $32$ and $24$ .

$35 \frac{d y}{d t} + 56 = 0$

Step 1.1.2

Subtract $56$ from both sides of the equation.

$35 \frac{d y}{d t} = - 56$

Step 1.1.3

Divide each term in $35 \frac{d y}{d t} = - 56$ by $35$ and simplify.

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Step 1.1.3.1

Divide each term in $35 \frac{d y}{d t} = - 56$ by $35$ .

$\frac{35 \frac{d y}{d t}}{35} = \frac{- 56}{35}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $35$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{35 \frac{d y}{d t}}{35} = \frac{- 56}{35}$

Step 1.1.3.2.1.2

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

$\frac{d y}{d t} = \frac{- 56}{35}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Cancel the common factor of $- 56$ and $35$ .

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Step 1.1.3.3.1.1

Factor $7$ out of $- 56$ .

$\frac{d y}{d t} = \frac{7 (- 8)}{35}$

Step 1.1.3.3.1.2

Cancel the common factors.

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Step 1.1.3.3.1.2.1

Factor $7$ out of $35$ .

$\frac{d y}{d t} = \frac{7 \cdot - 8}{7 \cdot 5}$

Step 1.1.3.3.1.2.2

Cancel the common factor.

$\frac{d y}{d t} = \frac{7 \cdot - 8}{7 \cdot 5}$

Step 1.1.3.3.1.2.3

Rewrite the expression.

$\frac{d y}{d t} = \frac{- 8}{5}$

Step 1.1.3.3.2

Move the negative in front of the fraction.

$\frac{d y}{d t} = - \frac{8}{5}$

Step 1.2

Rewrite the equation.

$d y = - \frac{8}{5} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - \frac{8}{5} d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - \frac{8}{5} d t$

Step 2.3

Apply the constant rule.

$y + C_{1} = - \frac{8}{5} t + C_{2}$

Step 2.4

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

$y = - \frac{8}{5} t + K$