Calculus Examples

$\frac{d y}{d t} = 2 \sqrt{t}$ , $y (1) = 4$

Step 1

Rewrite the equation.

$d y = 2 \sqrt{t} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 2 \sqrt{t} d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 2 \sqrt{t} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

$y + C_{1} = 2 \int \sqrt{t} d t$

Step 2.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$y + C_{1} = 2 \int t^{\frac{1}{2}} d t$

Step 2.3.3

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

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

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

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

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

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Combine $2$ and $\frac{2}{3}$ .

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

Step 2.3.4.2.2

Multiply $2$ by $2$ .

$y + C_{1} = \frac{4}{3} t^{\frac{3}{2}} + C_{2}$

Step 2.4

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

$y = \frac{4}{3} t^{\frac{3}{2}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $t$ and $4$ for $y$ in $y = \frac{4}{3} t^{\frac{3}{2}} + K$ .

$4 = \frac{4}{3} \cdot 1^{\frac{3}{2}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{4}{3} \cdot 1^{\frac{3}{2}} + K = 4$ .

$\frac{4}{3} \cdot 1^{\frac{3}{2}} + K = 4$

Step 4.2

Simplify each term.

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Step 4.2.1

One to any power is one.

$\frac{4}{3} \cdot 1 + K = 4$

Step 4.2.2

Multiply $\frac{4}{3}$ by $1$ .

$\frac{4}{3} + K = 4$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Subtract $\frac{4}{3}$ from both sides of the equation.

$K = 4 - \frac{4}{3}$

Step 4.3.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$K = 4 \cdot \frac{3}{3} - \frac{4}{3}$

Step 4.3.3

Combine $4$ and $\frac{3}{3}$ .

$K = \frac{4 \cdot 3}{3} - \frac{4}{3}$

Step 4.3.4

Combine the numerators over the common denominator.

$K = \frac{4 \cdot 3 - 4}{3}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $4$ by $3$ .

$K = \frac{12 - 4}{3}$

Step 4.3.5.2

Subtract $4$ from $12$ .

$K = \frac{8}{3}$

Step 5

Substitute $\frac{8}{3}$ for $K$ in $y = \frac{4}{3} t^{\frac{3}{2}} + K$ and simplify.

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Step 5.1

Substitute $\frac{8}{3}$ for $K$ .

$y = \frac{4}{3} t^{\frac{3}{2}} + \frac{8}{3}$

Step 5.2

Combine $\frac{4}{3}$ and $t^{\frac{3}{2}}$ .

$y = \frac{4 t^{\frac{3}{2}}}{3} + \frac{8}{3}$