Calculus Examples

$\frac{d y}{d t} + 3 y = 7$ , $y (0) = 1$

Step 1

The integrating factor is defined by the formula $e^{\int P (t) d t}$ , where $P (t) = 3$ .

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Step 1.1

Set up the integration.

$e^{\int 3 d t}$

Step 1.2

Apply the constant rule.

$e^{3 t + C}$

Step 1.3

Remove the constant of integration.

$e^{3 t}$

Step 2

Multiply each term by the integrating factor $e^{3 t}$ .

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Step 2.1

Multiply each term by $e^{3 t}$ .

$e^{3 t} \frac{d y}{d t} + e^{3 t} (3 y) = e^{3 t} \cdot 7$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{3 t} \frac{d y}{d t} + 3 e^{3 t} y = e^{3 t} \cdot 7$

Step 2.3

Move $7$ to the left of $e^{3 t}$ .

$e^{3 t} \frac{d y}{d t} + 3 e^{3 t} y = 7 \cdot e^{3 t}$

Step 2.4

Reorder factors in $e^{3 t} \frac{d y}{d t} + 3 e^{3 t} y = 7 e^{3 t}$ .

$e^{3 t} \frac{d y}{d t} + 3 y e^{3 t} = 7 e^{3 t}$

Step 3

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

$\frac{d}{d t} [e^{3 t} y] = 7 e^{3 t}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d t} [e^{3 t} y] d t = \int 7 e^{3 t} d t$

Step 5

Integrate the left side.

$e^{3 t} y = \int 7 e^{3 t} d t$

Step 6

Integrate the right side.

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Step 6.1

Since $7$ is constant with respect to $t$ , move $7$ out of the integral.

$e^{3 t} y = 7 \int e^{3 t} d t$

Step 6.2

Let $u = 3 t$ . Then $d u = 3 d t$ , so $\frac{1}{3} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.1

Let $u = 3 t$ . Find $\frac{d u}{d t}$ .

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Step 6.2.1.1

Differentiate $3 t$ .

$\frac{d}{d t} [3 t]$

Step 6.2.1.2

Since $3$ is constant with respect to $t$ , the derivative of $3 t$ with respect to $t$ is $3 \frac{d}{d t} [t]$ .

$3 \frac{d}{d t} [t]$

Step 6.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 6.2.1.4

Multiply $3$ by $1$ .

$3$

Step 6.2.2

Rewrite the problem using $u$ and $d u$ .

$e^{3 t} y = 7 \int e^{u} \frac{1}{3} d u$

Step 6.3

Combine $e^{u}$ and $\frac{1}{3}$ .

$e^{3 t} y = 7 \int \frac{e^{u}}{3} d u$

Step 6.4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$e^{3 t} y = 7 (\frac{1}{3} \int e^{u} d u)$

Step 6.5

Combine $\frac{1}{3}$ and $7$ .

$e^{3 t} y = \frac{7}{3} \int e^{u} d u$

Step 6.6

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{3 t} y = \frac{7}{3} (e^{u} + C)$

Step 6.7

Simplify.

$e^{3 t} y = \frac{7}{3} e^{u} + C$

Step 6.8

Replace all occurrences of $u$ with $3 t$ .

$e^{3 t} y = \frac{7}{3} e^{3 t} + C$

Step 7

Divide each term in $e^{3 t} y = \frac{7}{3} e^{3 t} + C$ by $e^{3 t}$ and simplify.

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Step 7.1

Divide each term in $e^{3 t} y = \frac{7}{3} e^{3 t} + C$ by $e^{3 t}$ .

$\frac{e^{3 t} y}{e^{3 t}} = \frac{\frac{7}{3} e^{3 t}}{e^{3 t}} + \frac{C}{e^{3 t}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{3 t} y}{e^{3 t}} = \frac{\frac{7}{3} e^{3 t}}{e^{3 t}} + \frac{C}{e^{3 t}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{7}{3} e^{3 t}}{e^{3 t}} + \frac{C}{e^{3 t}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

Cancel the common factor.

$y = \frac{\frac{7}{3} e^{3 t}}{e^{3 t}} + \frac{C}{e^{3 t}}$

Step 7.3.1.2

Divide $\frac{7}{3}$ by $1$ .

$y = \frac{7}{3} + \frac{C}{e^{3 t}}$

Step 8

Use the initial condition to find the value of $C$ by substituting $0$ for $t$ and $1$ for $y$ in $y = \frac{7}{3} + \frac{C}{e^{3 t}}$ .

$1 = \frac{7}{3} + \frac{C}{e^{3 \cdot 0}}$

Step 9

Solve for $C$ .

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Step 9.1

Rewrite the equation as $\frac{7}{3} + \frac{C}{e^{3 \cdot 0}} = 1$ .

$\frac{7}{3} + \frac{C}{e^{3 \cdot 0}} = 1$

Step 9.2

Simplify each term.

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Step 9.2.1

Simplify the denominator.

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Step 9.2.1.1

Multiply $3$ by $0$ .

$\frac{7}{3} + \frac{C}{e^{0}} = 1$

Step 9.2.1.2

Anything raised to $0$ is $1$ .

$\frac{7}{3} + \frac{C}{1} = 1$

Step 9.2.2

Divide $C$ by $1$ .

$\frac{7}{3} + C = 1$

Step 9.3

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

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Step 9.3.1

Subtract $\frac{7}{3}$ from both sides of the equation.

$C = 1 - \frac{7}{3}$

Step 9.3.2

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

$C = \frac{3}{3} - \frac{7}{3}$

Step 9.3.3

Combine the numerators over the common denominator.

$C = \frac{3 - 7}{3}$

Step 9.3.4

Subtract $7$ from $3$ .

$C = \frac{- 4}{3}$

Step 9.3.5

Move the negative in front of the fraction.

$C = - \frac{4}{3}$

Step 10

Substitute $- \frac{4}{3}$ for $C$ in $y = \frac{7}{3} + \frac{C}{e^{3 t}}$ and simplify.

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Step 10.1

Substitute $- \frac{4}{3}$ for $C$ .

$y = \frac{7}{3} + \frac{- \frac{4}{3}}{e^{3 t}}$

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{7}{3} - \frac{4}{3} \cdot \frac{1}{e^{3 t}}$

Step 10.2.2

Multiply $\frac{1}{e^{3 t}}$ by $\frac{4}{3}$ .

$y = \frac{7}{3} - \frac{4}{e^{3 t} \cdot 3}$

Step 10.2.3

Move $3$ to the left of $e^{3 t}$ .

$y = \frac{7}{3} - \frac{4}{3 e^{3 t}}$