Calculus Examples

$\frac{d y}{d x} = 3 x^{2}$ , $y (0) = 5$

Step 1

Rewrite the equation.

$d y = 3 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 3 x^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 3 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$y + C_{1} = 3 \int x^{2} d x$

Step 2.3.2

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

$y + C_{1} = 3 (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

$y + C_{1} = 3 (\frac{1}{3}) x^{3} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

$y + C_{1} = \frac{3}{3} x^{3} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $3$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

$y + C_{1} = \frac{3}{3} x^{3} + C_{2}$

Step 2.3.3.2.2.2

Rewrite the expression.

$y + C_{1} = 1 x^{3} + C_{2}$

Step 2.3.3.2.3

Multiply $x^{3}$ by $1$ .

$y + C_{1} = x^{3} + C_{2}$

Step 2.4

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

$y = x^{3} + K$

Step 3

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

$5 = 0^{3} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $0^{3} + K = 5$ .

$0^{3} + K = 5$

Step 4.2

Simplify $0^{3} + K$ .

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Step 4.2.1

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

$0 + K = 5$

Step 4.2.2

Add $0$ and $K$ .

$K = 5$

Step 5

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

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Step 5.1

Substitute $5$ for $K$ .

$y = x^{3} + 5$