Calculus Examples

$\frac{d z}{d t} - \frac{1}{2 t} z = 0$

Step 1

The integrating factor is defined by the formula $e^{\int P (t) d t}$ , where $P (t) = - \frac{1}{2 t}$ .

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Step 1.1

Set up the integration.

$e^{\int - \frac{1}{2 t} d t}$

Step 1.2

Integrate $- \frac{1}{2 t}$ .

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Step 1.2.1

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

$e^{- \int \frac{1}{2 t} d t}$

Step 1.2.2

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$e^{- (\frac{1}{2} \int \frac{1}{t} d t)}$

Step 1.2.3

The integral of $\frac{1}{t}$ with respect to $t$ is $\ln (| t |)$ .

$e^{- \frac{1}{2} (\ln (| t |) + C)}$

Step 1.2.4

Simplify.

$e^{- \frac{1}{2} \ln (| t |) + C}$

Step 1.3

Remove the constant of integration.

$e^{- \frac{1}{2} \ln (t)}$

Step 1.4

Use the logarithmic power rule.

$e^{\ln (t^{- \frac{1}{2}})}$

Step 1.5

Exponentiation and log are inverse functions.

$t^{- \frac{1}{2}}$

Step 1.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{t^{\frac{1}{2}}}$

Step 2

Multiply each term by the integrating factor $\frac{1}{t^{\frac{1}{2}}}$ .

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Step 2.1

Multiply each term by $\frac{1}{t^{\frac{1}{2}}}$ .

$\frac{1}{t^{\frac{1}{2}}} \frac{d z}{d t} + \frac{1}{t^{\frac{1}{2}}} (- \frac{1}{2 t} z) = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{1}{t^{\frac{1}{2}}}$ and $\frac{d z}{d t}$ .

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} + \frac{1}{t^{\frac{1}{2}}} (- \frac{1}{2 t} z) = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.2.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} - \frac{1}{t^{\frac{1}{2}}} (\frac{1}{2 t} z) = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.2.3

Combine $\frac{1}{2 t}$ and $z$ .

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} - \frac{1}{t^{\frac{1}{2}}} \cdot \frac{z}{2 t} = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.2.4

Multiply $- \frac{1}{t^{\frac{1}{2}}} \cdot \frac{z}{2 t}$ .

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Step 2.2.4.1

Multiply $\frac{z}{2 t}$ by $\frac{1}{t^{\frac{1}{2}}}$ .

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} - \frac{z}{2 t \cdot t^{\frac{1}{2}}} = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.2.4.2

Multiply $t$ by $t^{\frac{1}{2}}$ by adding the exponents.

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Step 2.2.4.2.1

Move $t^{\frac{1}{2}}$ .

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} - \frac{z}{2 (t^{\frac{1}{2}} t)} = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.2.4.2.2

Multiply $t^{\frac{1}{2}}$ by $t$ .

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Step 2.2.4.2.2.1

Raise $t$ to the power of $1$ .

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} - \frac{z}{2 (t^{\frac{1}{2}} t^{1})} = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.2.4.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} - \frac{z}{2 t^{\frac{1}{2} + 1}} = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.2.4.2.3

Write $1$ as a fraction with a common denominator.

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} - \frac{z}{2 t^{\frac{1}{2} + \frac{2}{2}}} = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.2.4.2.4

Combine the numerators over the common denominator.

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} - \frac{z}{2 t^{\frac{1 + 2}{2}}} = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.2.4.2.5

Add $1$ and $2$ .

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} - \frac{z}{2 t^{\frac{3}{2}}} = \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 2.3

Multiply $\frac{1}{t^{\frac{1}{2}}}$ by $0$ .

$\frac{\frac{d z}{d t}}{t^{\frac{1}{2}}} - \frac{z}{2 t^{\frac{3}{2}}} = 0$

Step 3

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d t} [\frac{1}{t^{\frac{1}{2}}} z] = 0$

Step 4

Set up an integral on each side.

$\int \frac{d}{d t} [\frac{1}{t^{\frac{1}{2}}} z] d t = \int 0 d t$

Step 5

Integrate the left side.

$\frac{1}{t^{\frac{1}{2}}} z = \int 0 d t$

Step 6

Integrate the right side.

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Step 6.1

The integral of $0$ with respect to $t$ is $0$ .

$\frac{1}{t^{\frac{1}{2}}} z = 0 + C$

Step 6.2

Add $0$ and $C$ .

$\frac{1}{t^{\frac{1}{2}}} z = C$

Step 7

Solve for $z$ .

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Step 7.1

Combine $\frac{1}{t^{\frac{1}{2}}}$ and $z$ .

$\frac{z}{t^{\frac{1}{2}}} = C$

Step 7.2

Multiply both sides by $t^{\frac{1}{2}}$ .

$\frac{z}{t^{\frac{1}{2}}} t^{\frac{1}{2}} = C t^{\frac{1}{2}}$

Step 7.3

Simplify the left side.

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Step 7.3.1

Cancel the common factor of $t^{\frac{1}{2}}$ .

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Step 7.3.1.1

Cancel the common factor.

$\frac{z}{t^{\frac{1}{2}}} t^{\frac{1}{2}} = C t^{\frac{1}{2}}$

Step 7.3.1.2

Rewrite the expression.

$z = C t^{\frac{1}{2}}$