Calculus Examples

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

Step 1

Write the problem as a mathematical expression.

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

Step 2

Separate the variables.

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Step 2.1

Multiply both sides by $y$ .

$y \frac{d y}{d t} = y \frac{3 t^{2}}{y}$

Step 2.2

Cancel the common factor of $y$ .

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Step 2.2.1

Cancel the common factor.

$y \frac{d y}{d t} = y \frac{3 t^{2}}{y}$

Step 2.2.2

Rewrite the expression.

$y \frac{d y}{d t} = 3 t^{2}$

Step 2.3

Rewrite the equation.

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

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

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

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

Step 3.3

Integrate the right side.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Simplify the answer.

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Step 3.3.3.1

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

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

Step 3.3.3.2

Simplify.

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Step 3.3.3.2.1

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

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

Step 3.3.3.2.2

Cancel the common factor of $3$ .

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Step 3.3.3.2.2.1

Cancel the common factor.

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

Step 3.3.3.2.2.2

Rewrite the expression.

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

Step 3.3.3.2.3

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

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

Step 3.4

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

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

Step 4

Solve for $y$ .

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Step 4.1

Multiply both sides of the equation by $2$ .

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

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 4.2.1.1.1

Combine $\frac{1}{2}$ and $y^{2}$ .

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

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (t^{3} + K)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Apply the distributive property.

$y^{2} = 2 t^{3} + 2 K$

Step 4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{2 t^{3} + 2 K}$

Step 4.4

Factor $2$ out of $2 t^{3} + 2 K$ .

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Step 4.4.1

Factor $2$ out of $2 t^{3}$ .

$y = \pm \sqrt{2 (t^{3}) + 2 K}$

Step 4.4.2

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 (t^{3}) + 2 K}$

Step 4.4.3

Factor $2$ out of $2 (t^{3}) + 2 K$ .

$y = \pm \sqrt{2 (t^{3} + K)}$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{2 (t^{3} + K)}$

Step 4.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{2 (t^{3} + K)}$

Step 4.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{2 (t^{3} + K)}$

$y = - \sqrt{2 (t^{3} + K)}$

$y = \sqrt{2 (t^{3} + K)}$

$y = - \sqrt{2 (t^{3} + K)}$

$y = \sqrt{2 (t^{3} + K)}$

$y = - \sqrt{2 (t^{3} + K)}$

Step 5

Since $y$ is non-negative in the initial condition $(2, 0)$ , only consider $y = \sqrt{2 (t^{3} + K)}$ to find the $K$ . Substitute $2$ for $t$ and $0$ for $y$ .

$0 = \sqrt{2 (2^{3} + K)}$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $\sqrt{2 (2^{3} + K)} = 0$ .

$\sqrt{2 (2^{3} + K)} = 0$

Step 6.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{2 (2^{3} + K)}}^{2} = 0^{2}$

Step 6.3

Simplify each side of the equation.

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Step 6.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (2^{3} + K)}$ as ${(2 (2^{3} + K))}^{\frac{1}{2}}$ .

${({(2 (2^{3} + K))}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Simplify ${({(2 (2^{3} + K))}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.1.1

Simplify the expression.

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Step 6.3.2.1.1.1

Multiply the exponents in ${({(2 (2^{3} + K))}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 (2^{3} + K))}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.1.1.1.2.1

Cancel the common factor.

${(2 (2^{3} + K))}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.1.1.1.2.2

Rewrite the expression.

${(2 (2^{3} + K))}^{1} = 0^{2}$

Step 6.3.2.1.1.2

Raise $2$ to the power of $3$ .

${(2 (8 + K))}^{1} = 0^{2}$

Step 6.3.2.1.2

Apply the distributive property.

${(2 \cdot 8 + 2 K)}^{1} = 0^{2}$

Step 6.3.2.1.3

Multiply.

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Step 6.3.2.1.3.1

Multiply $2$ by $8$ .

${(16 + 2 K)}^{1} = 0^{2}$

Step 6.3.2.1.3.2

Simplify.

$16 + 2 K = 0^{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

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

$16 + 2 K = 0$

Step 6.4

Solve for $K$ .

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Step 6.4.1

Subtract $16$ from both sides of the equation.

$2 K = - 16$

Step 6.4.2

Divide each term in $2 K = - 16$ by $2$ and simplify.

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Step 6.4.2.1

Divide each term in $2 K = - 16$ by $2$ .

$\frac{2 K}{2} = \frac{- 16}{2}$

Step 6.4.2.2

Simplify the left side.

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Step 6.4.2.2.1

Cancel the common factor of $2$ .

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Step 6.4.2.2.1.1

Cancel the common factor.

$\frac{2 K}{2} = \frac{- 16}{2}$

Step 6.4.2.2.1.2

Divide $K$ by $1$ .

$K = \frac{- 16}{2}$

Step 6.4.2.3

Simplify the right side.

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Step 6.4.2.3.1

Divide $- 16$ by $2$ .

$K = - 8$

Step 7

Substitute $- 8$ for $K$ in $y = \sqrt{2 (t^{3} + K)}$ and simplify.

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Step 7.1

Substitute $- 8$ for $K$ .

$y = \sqrt{2 (t^{3} - 8)}$

Step 7.2

Rewrite $8$ as $2^{3}$ .

$y = \sqrt{2 (t^{3} - 2^{3})}$

Step 7.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = t$ and $b = 2$ .

$y = \sqrt{2 ((t - 2) (t^{2} + t \cdot 2 + 2^{2}))}$

Step 7.4

Simplify.

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Step 7.4.1

Move $2$ to the left of $t$ .

$y = \sqrt{2 ((t - 2) (t^{2} + 2 \cdot t + 2^{2}))}$

Step 7.4.2

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

$y = \sqrt{2 ((t - 2) (t^{2} + 2 t + 4))}$

$y = \sqrt{2 (t - 2) (t^{2} + 2 t + 4)}$