Calculus Examples

$\frac{d y}{d t} - \frac{5}{t} \cdot y = t^{5}$

Step 1

The integrating factor is defined by the formula $e^{\int P (t) d t}$ , where $P (t) = - \frac{5}{t}$ .

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Step 1.1

Set up the integration.

$e^{\int - \frac{5}{t} d t}$

Step 1.2

Integrate $- \frac{5}{t}$ .

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Step 1.2.1

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

$e^{- \int \frac{5}{t} d t}$

Step 1.2.2

Since $5$ is constant with respect to $t$ , move $5$ out of the integral.

$e^{- (5 \int \frac{1}{t} d t)}$

Step 1.2.3

Multiply $5$ by $- 1$ .

$e^{- 5 \int \frac{1}{t} d t}$

Step 1.2.4

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

$e^{- 5 (\ln (| t |) + C)}$

Step 1.2.5

Simplify.

$e^{- 5 \ln (| t |) + C}$

Step 1.3

Remove the constant of integration.

$e^{- 5 \ln (t)}$

Step 1.4

Use the logarithmic power rule.

$e^{\ln (t^{- 5})}$

Step 1.5

Exponentiation and log are inverse functions.

$t^{- 5}$

Step 1.6

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

$\frac{1}{t^{5}}$

Step 2

Multiply each term by the integrating factor $\frac{1}{t^{5}}$ .

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Step 2.1

Multiply each term by $\frac{1}{t^{5}}$ .

$\frac{1}{t^{5}} \frac{d y}{d t} + \frac{1}{t^{5}} (- \frac{5}{t} \cdot y) = \frac{1}{t^{5}} t^{5}$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{1}{t^{5}}$ and $\frac{d y}{d t}$ .

$\frac{\frac{d y}{d t}}{t^{5}} + \frac{1}{t^{5}} (- \frac{5}{t} \cdot y) = \frac{1}{t^{5}} t^{5}$

Step 2.2.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d t}}{t^{5}} - \frac{1}{t^{5}} (\frac{5}{t} y) = \frac{1}{t^{5}} t^{5}$

Step 2.2.3

Combine $\frac{5}{t}$ and $y$ .

$\frac{\frac{d y}{d t}}{t^{5}} - \frac{1}{t^{5}} \cdot \frac{5 y}{t} = \frac{1}{t^{5}} t^{5}$

Step 2.2.4

Multiply $- \frac{1}{t^{5}} \cdot \frac{5 y}{t}$ .

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Step 2.2.4.1

Multiply $\frac{5 y}{t}$ by $\frac{1}{t^{5}}$ .

$\frac{\frac{d y}{d t}}{t^{5}} - \frac{5 y}{t \cdot t^{5}} = \frac{1}{t^{5}} t^{5}$

Step 2.2.4.2

Multiply $t$ by $t^{5}$ by adding the exponents.

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Step 2.2.4.2.1

Multiply $t$ by $t^{5}$ .

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Step 2.2.4.2.1.1

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

$\frac{\frac{d y}{d t}}{t^{5}} - \frac{5 y}{t^{1} t^{5}} = \frac{1}{t^{5}} t^{5}$

Step 2.2.4.2.1.2

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

$\frac{\frac{d y}{d t}}{t^{5}} - \frac{5 y}{t^{1 + 5}} = \frac{1}{t^{5}} t^{5}$

Step 2.2.4.2.2

Add $1$ and $5$ .

$\frac{\frac{d y}{d t}}{t^{5}} - \frac{5 y}{t^{6}} = \frac{1}{t^{5}} t^{5}$

Step 2.3

Cancel the common factor of $t^{5}$ .

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Step 2.3.1

Cancel the common factor.

$\frac{\frac{d y}{d t}}{t^{5}} - \frac{5 y}{t^{6}} = \frac{1}{t^{5}} t^{5}$

Step 2.3.2

Rewrite the expression.

$\frac{\frac{d y}{d t}}{t^{5}} - \frac{5 y}{t^{6}} = 1$

Step 3

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

$\frac{d}{d t} [\frac{1}{t^{5}} y] = 1$

Step 4

Set up an integral on each side.

$\int \frac{d}{d t} [\frac{1}{t^{5}} y] d t = \int d t$

Step 5

Integrate the left side.

$\frac{1}{t^{5}} y = \int d t$

Step 6

Apply the constant rule.

$\frac{1}{t^{5}} y = t + C$

Step 7

Solve for $y$ .

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Step 7.1

Combine $\frac{1}{t^{5}}$ and $y$ .

$\frac{y}{t^{5}} = t + C$

Step 7.2

Multiply both sides by $t^{5}$ .

$\frac{y}{t^{5}} t^{5} = (t + C) t^{5}$

Step 7.3

Simplify.

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Step 7.3.1

Simplify the left side.

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Step 7.3.1.1

Cancel the common factor of $t^{5}$ .

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Step 7.3.1.1.1

Cancel the common factor.

$\frac{y}{t^{5}} t^{5} = (t + C) t^{5}$

Step 7.3.1.1.2

Rewrite the expression.

$y = (t + C) t^{5}$

Step 7.3.2

Simplify the right side.

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Step 7.3.2.1

Simplify $(t + C) t^{5}$ .

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Step 7.3.2.1.1

Apply the distributive property.

$y = t \cdot t^{5} + C t^{5}$

Step 7.3.2.1.2

Multiply $t$ by $t^{5}$ by adding the exponents.

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Step 7.3.2.1.2.1

Multiply $t$ by $t^{5}$ .

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Step 7.3.2.1.2.1.1

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

$y = t^{1} t^{5} + C t^{5}$

Step 7.3.2.1.2.1.2

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

$y = t^{1 + 5} + C t^{5}$

Step 7.3.2.1.2.2

Add $1$ and $5$ .

$y = t^{6} + C t^{5}$

Step 7.3.2.1.3

Reorder $t^{6}$ and $C t^{5}$ .

$y = C t^{5} + t^{6}$