Calculus Examples

$\frac{d y}{d t} + \frac{y}{t} = 3 t$ , $y (2) = 8$

Step 1

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

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Step 1.1

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

$\frac{d y}{d t} + y \frac{1}{t} = 3 t$

Step 1.2

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

$\frac{d y}{d t} + \frac{1}{t} y = 3 t$

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

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

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

Step 2.3

Remove the constant of integration.

$e^{\ln (t)}$

Step 2.4

Exponentiation and log are inverse functions.

$t$

Step 3

Multiply each term by the integrating factor $t$ .

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Step 3.1

Multiply each term by $t$ .

$t \frac{d y}{d t} + t (\frac{1}{t} y) = t (3 t)$

Step 3.2

Simplify each term.

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Step 3.2.1

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

$t \frac{d y}{d t} + t \frac{y}{t} = t (3 t)$

Step 3.2.2

Cancel the common factor of $t$ .

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Step 3.2.2.1

Cancel the common factor.

$t \frac{d y}{d t} + t \frac{y}{t} = t (3 t)$

Step 3.2.2.2

Rewrite the expression.

$t \frac{d y}{d t} + y = t (3 t)$

Step 3.3

Rewrite using the commutative property of multiplication.

$t \frac{d y}{d t} + y = 3 t \cdot t$

Step 3.4

Multiply $t$ by $t$ by adding the exponents.

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Step 3.4.1

Move $t$ .

$t \frac{d y}{d t} + y = 3 (t \cdot t)$

Step 3.4.2

Multiply $t$ by $t$ .

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify the answer.

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Step 7.3.1

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

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

Step 7.3.2

Simplify.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Cancel the common factor of $3$ .

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Step 7.3.2.2.1

Cancel the common factor.

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

Step 7.3.2.2.2

Rewrite the expression.

$t y = 1 t^{3} + C$

Step 7.3.2.3

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

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

Step 8

Divide each term in $t y = t^{3} + C$ by $t$ and simplify.

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Step 8.1

Divide each term in $t y = t^{3} + C$ by $t$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $t$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

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

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Step 8.3.1.1

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

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

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

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

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

Step 8.3.1.2.2

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

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

Step 8.3.1.2.3

Cancel the common factor.

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

Step 8.3.1.2.4

Rewrite the expression.

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

Step 8.3.1.2.5

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

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

Step 9

Use the initial condition to find the value of $C$ by substituting $2$ for $t$ and $8$ for $y$ in $y = t^{2} + \frac{C}{t}$ .

$8 = 2^{2} + \frac{C}{2}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $2^{2} + \frac{C}{2} = 8$ .

$2^{2} + \frac{C}{2} = 8$

Step 10.2

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

$4 + \frac{C}{2} = 8$

Step 10.3

Move all terms not containing $C$ to the right side of the equation.

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Step 10.3.1

Subtract $4$ from both sides of the equation.

$\frac{C}{2} = 8 - 4$

Step 10.3.2

Subtract $4$ from $8$ .

$\frac{C}{2} = 4$

Step 10.4

Multiply both sides of the equation by $2$ .

$2 \frac{C}{2} = 2 \cdot 4$

Step 10.5

Simplify both sides of the equation.

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Step 10.5.1

Simplify the left side.

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Step 10.5.1.1

Cancel the common factor of $2$ .

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Step 10.5.1.1.1

Cancel the common factor.

$2 \frac{C}{2} = 2 \cdot 4$

Step 10.5.1.1.2

Rewrite the expression.

$C = 2 \cdot 4$

Step 10.5.2

Simplify the right side.

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Step 10.5.2.1

Multiply $2$ by $4$ .

$C = 8$

Step 11

Substitute $8$ for $C$ in $y = t^{2} + \frac{C}{t}$ and simplify.

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Step 11.1

Substitute $8$ for $C$ .

$y = t^{2} + \frac{8}{t}$