Calculus Examples

$\frac{d Y}{d x} = - 15 x^{2}$ , $Y (0) = 6$

Step 1

Rewrite the equation.

$d Y = - 15 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d Y = \int - 15 x^{2} d x$

Step 2.2

Apply the constant rule.

$Y + C_{1} = \int - 15 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 15$ is constant with respect to $x$ , move $- 15$ out of the integral.

$Y + C_{1} = - 15 \int x^{2} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$Y + C_{1} = - 15 (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $- 15 (\frac{1}{3} x^{3} + C_{2})$ as $- 15 (\frac{1}{3}) x^{3} + C_{2}$ .

$Y + C_{1} = - 15 (\frac{1}{3}) x^{3} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $- 15$ and $\frac{1}{3}$ .

$Y + C_{1} = \frac{- 15}{3} x^{3} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $- 15$ and $3$ .

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Step 2.3.3.2.2.1

Factor $3$ out of $- 15$ .

$Y + C_{1} = \frac{3 \cdot - 5}{3} x^{3} + C_{2}$

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $3$ out of $3$ .

$Y + C_{1} = \frac{3 \cdot - 5}{3 (1)} x^{3} + C_{2}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$Y + C_{1} = \frac{3 \cdot - 5}{3 \cdot 1} x^{3} + C_{2}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$Y + C_{1} = \frac{- 5}{1} x^{3} + C_{2}$

Step 2.3.3.2.2.2.4

Divide $- 5$ by $1$ .

$Y + C_{1} = - 5 x^{3} + C_{2}$

Step 2.4

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

$Y = - 5 x^{3} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $6$ for $Y$ in $Y = - 5 x^{3} + K$ .

$6 = - 5 \cdot 0^{3} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- 5 \cdot 0^{3} + K = 6$ .

$- 5 \cdot 0^{3} + K = 6$

Step 4.2

Simplify $- 5 \cdot 0^{3} + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$- 5 \cdot 0 + K = 6$

Step 4.2.1.2

Multiply $- 5$ by $0$ .

$0 + K = 6$

Step 4.2.2

Add $0$ and $K$ .

$K = 6$

Step 5

Substitute $6$ for $K$ in $Y = - 5 x^{3} + K$ and simplify.

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Step 5.1

Substitute $6$ for $K$ .

$Y = - 5 x^{3} + 6$