Calculus Examples

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

Step 1

Rewrite the differential equation as $\frac{d y}{d t} + M (t) y = Q (t)$ .

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Step 1.1

Divide each term in $t \frac{d y}{d t} - y = 2 t^{3}$ by $t$ .

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

Step 1.2

Cancel the common factor of $t$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Divide $\frac{d y}{d t}$ by $1$ .

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

Step 1.3

Cancel the common factor of $t^{3}$ and $t$ .

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Step 1.3.1

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

$\frac{d y}{d t} + \frac{- y}{t} = \frac{t (2 t^{2})}{t}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

$\frac{d y}{d t} + \frac{- y}{t} = \frac{t (2 t^{2})}{t^{1}}$

Step 1.3.2.2

Factor $t$ out of $t^{1}$ .

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

Step 1.3.2.3

Cancel the common factor.

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

Step 1.3.2.4

Rewrite the expression.

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

Step 1.3.2.5

Divide $2 t^{2}$ by $1$ .

$\frac{d y}{d t} + \frac{- y}{t} = 2 t^{2}$

Step 1.4

Factor $y$ out of $\frac{- y}{t}$ .

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

Step 1.5

Reorder $y$ and $\frac{- 1}{t}$ .

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

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

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Step 2.2.1

Split the fraction into multiple fractions.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Simplify.

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

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

Step 2.5

Exponentiation and log are inverse functions.

$t^{- 1}$

Step 2.6

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

$\frac{1}{t}$

Step 3

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

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Step 3.1

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

$\frac{1}{t} \frac{d y}{d t} + \frac{1}{t} (\frac{- 1}{t} y) = \frac{1}{t} (2 t^{2})$

Step 3.2

Simplify each term.

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Step 3.2.1

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

$\frac{\frac{d y}{d t}}{t} + \frac{1}{t} (\frac{- 1}{t} y) = \frac{1}{t} (2 t^{2})$

Step 3.2.2

Move the negative in front of the fraction.

$\frac{\frac{d y}{d t}}{t} + \frac{1}{t} (- \frac{1}{t} y) = \frac{1}{t} (2 t^{2})$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d t}}{t} - \frac{1}{t} (\frac{1}{t} y) = \frac{1}{t} (2 t^{2})$

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

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

$\frac{\frac{d y}{d t}}{t} - \frac{y}{t^{1} t} = \frac{1}{t} (2 t^{2})$

Step 3.2.5.3

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

$\frac{\frac{d y}{d t}}{t} - \frac{y}{t^{1} t^{1}} = \frac{1}{t} (2 t^{2})$

Step 3.2.5.4

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

$\frac{\frac{d y}{d t}}{t} - \frac{y}{t^{1 + 1}} = \frac{1}{t} (2 t^{2})$

Step 3.2.5.5

Add $1$ and $1$ .

$\frac{\frac{d y}{d t}}{t} - \frac{y}{t^{2}} = \frac{1}{t} (2 t^{2})$

Step 3.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d t}}{t} - \frac{y}{t^{2}} = 2 \frac{1}{t} t^{2}$

Step 3.4

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

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

Step 3.5

Cancel the common factor of $t$ .

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Step 3.5.1

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

$\frac{\frac{d y}{d t}}{t} - \frac{y}{t^{2}} = \frac{2}{t} (t \cdot t)$

Step 3.5.2

Cancel the common factor.

$\frac{\frac{d y}{d t}}{t} - \frac{y}{t^{2}} = \frac{2}{t} (t \cdot t)$

Step 3.5.3

Rewrite the expression.

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

Step 4

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

$\frac{d}{d t} [\frac{1}{t} y] = 2 t$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [\frac{1}{t} y] d t = \int 2 t d t$

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify the answer.

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Step 7.3.1

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

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

Step 7.3.2

Simplify.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Cancel the common factor of $2$ .

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Step 7.3.2.2.1

Cancel the common factor.

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

Step 7.3.2.2.2

Rewrite the expression.

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

Step 7.3.2.3

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

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

Step 8

Solve for $y$ .

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Step 8.1

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

$\frac{y}{t} = t^{2} + C$

Step 8.2

Multiply both sides by $t$ .

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Cancel the common factor of $t$ .

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

Rewrite the expression.

$y = (t^{2} + C) t$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $(t^{2} + C) t$ .

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Step 8.3.2.1.1

Apply the distributive property.

$y = t^{2} t + C t$

Step 8.3.2.1.2

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

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Step 8.3.2.1.2.1

Multiply $t^{2}$ by $t$ .

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Step 8.3.2.1.2.1.1

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

$y = t^{2} t^{1} + C t$

Step 8.3.2.1.2.1.2

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

$y = t^{2 + 1} + C t$

Step 8.3.2.1.2.2

Add $2$ and $1$ .

$y = t^{3} + C t$