Calculus Examples

$\frac{d y}{d t} = - b y (t)$

Step 1

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

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Step 1.1

Add $b y t$ to both sides of the equation.

$\frac{d y}{d t} + b y t = 0$

Step 1.2

Reorder terms.

$\frac{d y}{d t} + b t y = 0$

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int b t d t}$

Step 2.2

Integrate $b t$ .

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Step 2.2.1

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

$e^{b \int t d t}$

Step 2.2.2

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

$e^{b (\frac{1}{2} t^{2} + C)}$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

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

$e^{b \frac{1}{2} t^{2} + C}$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

$e^{\frac{b}{2} t^{2} + C}$

Step 2.2.3.2.2

Combine $\frac{b}{2}$ and $t^{2}$ .

$e^{\frac{b t^{2}}{2} + C}$

$e^{\frac{1}{2} b t^{2} + C}$

Step 2.3

Remove the constant of integration.

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

Step 2.4

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

$e^{\frac{b}{2} t^{2}}$

Step 2.5

Combine $\frac{b}{2}$ and $t^{2}$ .

$e^{\frac{b t^{2}}{2}}$

Step 3

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

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Step 3.1

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

$e^{\frac{b t^{2}}{2}} \frac{d y}{d t} + e^{\frac{b t^{2}}{2}} (b t y) = e^{\frac{b t^{2}}{2}} \cdot 0$

Step 3.2

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

$e^{\frac{b t^{2}}{2}} \frac{d y}{d t} + e^{\frac{b t^{2}}{2}} (b t y) = 0$

Step 3.3

Reorder factors in $e^{\frac{b t^{2}}{2}} \frac{d y}{d t} + e^{\frac{b t^{2}}{2}} (b t y) = 0$ .

$e^{\frac{b t^{2}}{2}} \frac{d y}{d t} + (b t) y e^{\frac{b t^{2}}{2}} = 0$

Step 4

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

$\frac{d}{d t} [e^{\frac{b t^{2}}{2}} y] = 0$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [e^{\frac{b t^{2}}{2}} y] d t = \int 0 d t$

Step 6

Integrate the left side.

$e^{\frac{b t^{2}}{2}} y = \int 0 d t$

Step 7

Integrate the right side.

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Step 7.1

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

$e^{\frac{b t^{2}}{2}} y = 0 + C$

Step 7.2

Add $0$ and $C$ .

$e^{\frac{b t^{2}}{2}} y = C$

Step 8

Divide each term in $e^{\frac{b t^{2}}{2}} y = C$ by $e^{\frac{b t^{2}}{2}}$ and simplify.

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Step 8.1

Divide each term in $e^{\frac{b t^{2}}{2}} y = C$ by $e^{\frac{b t^{2}}{2}}$ .

$\frac{e^{\frac{b t^{2}}{2}} y}{e^{\frac{b t^{2}}{2}}} = \frac{C}{e^{\frac{b t^{2}}{2}}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

$\frac{e^{\frac{b t^{2}}{2}} y}{e^{\frac{b t^{2}}{2}}} = \frac{C}{e^{\frac{b t^{2}}{2}}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{C}{e^{\frac{b t^{2}}{2}}}$