Calculus Examples

$\frac{d y}{d t} + 50 y = 0$

Step 1

The integrating factor is defined by the formula $e^{\int P (t) d t}$ , where $P (t) = 50$ .

Tap for more steps...

Step 1.1

Set up the integration.

$e^{\int 50 d t}$

Step 1.2

Apply the constant rule.

$e^{50 t + C}$

Step 1.3

Remove the constant of integration.

$e^{50 t}$

Step 2

Multiply each term by the integrating factor $e^{50 t}$ .

Tap for more steps...

Step 2.1

Multiply each term by $e^{50 t}$ .

$e^{50 t} \frac{d y}{d t} + e^{50 t} (50 y) = e^{50 t} \cdot 0$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{50 t} \frac{d y}{d t} + 50 e^{50 t} y = e^{50 t} \cdot 0$

Step 2.3

Multiply $e^{50 t}$ by $0$ .

$e^{50 t} \frac{d y}{d t} + 50 e^{50 t} y = 0$

Step 2.4

Reorder factors in $e^{50 t} \frac{d y}{d t} + 50 e^{50 t} y = 0$ .

$e^{50 t} \frac{d y}{d t} + 50 y e^{50 t} = 0$

Step 3

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

$\frac{d}{d t} [e^{50 t} y] = 0$

Step 4

Set up an integral on each side.

$\int \frac{d}{d t} [e^{50 t} y] d t = \int 0 d t$

Step 5

Integrate the left side.

$e^{50 t} y = \int 0 d t$

Step 6

Integrate the right side.

Tap for more steps...

Step 6.1

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

$e^{50 t} y = 0 + C$

Step 6.2

Add $0$ and $C$ .

$e^{50 t} y = C$

Step 7

Divide each term in $e^{50 t} y = C$ by $e^{50 t}$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $e^{50 t} y = C$ by $e^{50 t}$ .

$\frac{e^{50 t} y}{e^{50 t}} = \frac{C}{e^{50 t}}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $e^{50 t}$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{e^{50 t} y}{e^{50 t}} = \frac{C}{e^{50 t}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{C}{e^{50 t}}$