Calculus Examples

$\frac{d x}{d t} = \frac{9}{x}$ , $x (0) = 9$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $x$ .

$x \frac{d x}{d t} = x \frac{9}{x}$

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

$x \frac{d x}{d t} = x \frac{9}{x}$

Step 1.2.2

Rewrite the expression.

$x \frac{d x}{d t} = 9$

Step 1.3

Rewrite the equation.

$x d x = 9 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int x d x = \int 9 d t$

Step 2.2

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

$\frac{1}{2} x^{2} + C_{1} = \int 9 d t$

Step 2.3

Apply the constant rule.

$\frac{1}{2} x^{2} + C_{1} = 9 t + C_{2}$

Step 2.4

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

$\frac{1}{2} x^{2} = 9 t + K$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} x^{2}) = 2 (9 t + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

$2 \frac{x^{2}}{2} = 2 (9 t + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{x^{2}}{2} = 2 (9 t + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$x^{2} = 2 (9 t + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (9 t + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$x^{2} = 2 (9 t) + 2 K$

Step 3.2.2.1.2

Multiply $9$ by $2$ .

$x^{2} = 18 t + 2 K$

Step 3.3

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

$x = \pm \sqrt{18 t + 2 K}$

Step 3.4

Factor $2$ out of $18 t + 2 K$ .

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Step 3.4.1

Factor $2$ out of $18 t$ .

$x = \pm \sqrt{2 (9 t) + 2 K}$

Step 3.4.2

Factor $2$ out of $2 K$ .

$x = \pm \sqrt{2 (9 t) + 2 K}$

Step 3.4.3

Factor $2$ out of $2 (9 t) + 2 K$ .

$x = \pm \sqrt{2 (9 t + K)}$

Step 3.5

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

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Step 3.5.1

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

$x = \sqrt{2 (9 t + K)}$

Step 3.5.2

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

$x = - \sqrt{2 (9 t + K)}$

Step 3.5.3

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

$x = \sqrt{2 (9 t + K)}$

$x = - \sqrt{2 (9 t + K)}$

$x = \sqrt{2 (9 t + K)}$

$x = - \sqrt{2 (9 t + K)}$

$x = \sqrt{2 (9 t + K)}$

$x = - \sqrt{2 (9 t + K)}$

Step 4

Since $x$ is positive in the initial condition $(0, 9)$ , only consider $x = \sqrt{2 (9 t + K)}$ to find the $K$ . Substitute $0$ for $t$ and $9$ for $x$ .

$9 = \sqrt{2 (9 \cdot 0 + K)}$

Step 5

Solve for $K$ .

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Step 5.1

Rewrite the equation as $\sqrt{2 (9 \cdot 0 + K)} = 9$ .

$\sqrt{2 (9 \cdot 0 + K)} = 9$

Step 5.2

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

${\sqrt{2 (9 \cdot 0 + K)}}^{2} = 9^{2}$

Step 5.3

Simplify each side of the equation.

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Step 5.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (9 \cdot 0 + K)}$ as ${(2 (9 \cdot 0 + K))}^{\frac{1}{2}}$ .

${({(2 (9 \cdot 0 + K))}^{\frac{1}{2}})}^{2} = 9^{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify ${({(2 (9 \cdot 0 + K))}^{\frac{1}{2}})}^{2}$ .

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Step 5.3.2.1.1

Multiply the exponents in ${({(2 (9 \cdot 0 + K))}^{\frac{1}{2}})}^{2}$ .

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Step 5.3.2.1.1.1

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

${(2 (9 \cdot 0 + K))}^{\frac{1}{2} \cdot 2} = 9^{2}$

Step 5.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.3.2.1.1.2.1

Cancel the common factor.

${(2 (9 \cdot 0 + K))}^{\frac{1}{2} \cdot 2} = 9^{2}$

Step 5.3.2.1.1.2.2

Rewrite the expression.

${(2 (9 \cdot 0 + K))}^{1} = 9^{2}$

Step 5.3.2.1.2

Multiply $9$ by $0$ .

${(2 (0 + K))}^{1} = 9^{2}$

Step 5.3.2.1.3

Add $0$ and $K$ .

${(2 K)}^{1} = 9^{2}$

Step 5.3.2.1.4

Simplify.

$2 K = 9^{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Raise $9$ to the power of $2$ .

$2 K = 81$

Step 5.4

Divide each term in $2 K = 81$ by $2$ and simplify.

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Step 5.4.1

Divide each term in $2 K = 81$ by $2$ .

$\frac{2 K}{2} = \frac{81}{2}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{2 K}{2} = \frac{81}{2}$

Step 5.4.2.1.2

Divide $K$ by $1$ .

$K = \frac{81}{2}$

Step 6

Substitute $\frac{81}{2}$ for $K$ in $x = \sqrt{2 (9 t + K)}$ and simplify.

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Step 6.1

Substitute $\frac{81}{2}$ for $K$ .

$x = \sqrt{2 (9 t + \frac{81}{2})}$

Step 6.2

Factor $9$ out of $9 t + \frac{81}{2}$ .

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Step 6.2.1

Factor $9$ out of $9 t$ .

$x = \sqrt{2 (9 (t) + \frac{81}{2})}$

Step 6.2.2

Factor $9$ out of $\frac{81}{2}$ .

$x = \sqrt{2 (9 t + 9 (\frac{9}{2}))}$

Step 6.2.3

Factor $9$ out of $9 t + 9 (\frac{9}{2})$ .

$x = \sqrt{2 (9 (t + \frac{9}{2}))}$

$x = \sqrt{2 \cdot 9 (t + \frac{9}{2})}$

Step 6.3

Multiply $2$ by $9$ .

$x = \sqrt{18 (t + \frac{9}{2})}$

Step 6.4

To write $t$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \sqrt{18 (t \cdot \frac{2}{2} + \frac{9}{2})}$

Step 6.5

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

$x = \sqrt{18 (\frac{t \cdot 2}{2} + \frac{9}{2})}$

Step 6.6

Combine the numerators over the common denominator.

$x = \sqrt{18 \frac{t \cdot 2 + 9}{2}}$

Step 6.7

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

$x = \sqrt{18 \frac{2 t + 9}{2}}$

Step 6.8

Combine $18$ and $\frac{2 t + 9}{2}$ .

$x = \sqrt{\frac{18 (2 t + 9)}{2}}$

Step 6.9

Reduce the expression by cancelling the common factors.

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Step 6.9.1

Reduce the expression $\frac{18 (2 t + 9)}{2}$ by cancelling the common factors.

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Step 6.9.1.1

Factor $2$ out of $18 (2 t + 9)$ .

$x = \sqrt{\frac{2 (9 (2 t + 9))}{2}}$

Step 6.9.1.2

Factor $2$ out of $2$ .

$x = \sqrt{\frac{2 (9 (2 t + 9))}{2 (1)}}$

Step 6.9.1.3

Cancel the common factor.

$x = \sqrt{\frac{2 (9 (2 t + 9))}{2 \cdot 1}}$

Step 6.9.1.4

Rewrite the expression.

$x = \sqrt{\frac{9 (2 t + 9)}{1}}$

Step 6.9.2

Divide $9 (2 t + 9)$ by $1$ .

$x = \sqrt{9 (2 t + 9)}$

Step 6.10

Rewrite $9 (2 t + 9)$ as $3^{2} (2 t + 3^{2})$ .

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Step 6.10.1

Rewrite $9$ as $3^{2}$ .

$x = \sqrt{3^{2} (2 t + 9)}$

Step 6.10.2

Rewrite $9$ as $3^{2}$ .

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

Step 6.11

Pull terms out from under the radical.

$x = 3 \sqrt{2 t + 3^{2}}$

Step 6.12

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

$x = 3 \sqrt{2 t + 9}$