Calculus Examples

$e^{5 x} \frac{d y}{d x} + 5 e^{5 x} y = \sin (8 x)$

Step 1

Check if the left side of the equation is the result of the derivative of the term $e^{5 x} y$ .

Tap for more steps...

Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{5 x}$ and $g (x) = y$ .

$e^{5 x} \frac{d}{d x} [y] + y \frac{d}{d x} [e^{5 x}]$

Step 1.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$e^{5 x} y' + y \frac{d}{d x} [e^{5 x}]$

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 5 x$ .

Tap for more steps...

Step 1.3.1

To apply the Chain Rule, set $u$ as $5 x$ .

$e^{5 x} y' + y (\frac{d}{d u} [e^{u}] \frac{d}{d x} [5 x])$

Step 1.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{5 x} y' + y (e^{u} \frac{d}{d x} [5 x])$

Step 1.3.3

Replace all occurrences of $u$ with $5 x$ .

$e^{5 x} y' + y (e^{5 x} \frac{d}{d x} [5 x])$

Step 1.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$e^{5 x} y' + y (e^{5 x} (5 \frac{d}{d x} [x]))$

Step 1.5

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

$e^{5 x} y' + y (e^{5 x} (5 \cdot 1))$

Step 1.6

Multiply $5$ by $1$ .

$e^{5 x} y' + y (e^{5 x} \cdot 5)$

Step 1.7

Move $5$ to the left of $e^{5 x}$ .

$e^{5 x} y' + y (5 e^{5 x})$

Step 1.8

Substitute $\frac{d y}{d x}$ for $y'$ .

$e^{5 x} \frac{d y}{d x} + y (5 e^{5 x})$

Step 1.9

Remove parentheses.

$e^{5 x} \frac{d y}{d x} + y \cdot 5 e^{5 x}$

Step 1.10

Reorder $y$ and $5$ .

$e^{5 x} \frac{d y}{d x} + 5 \cdot y e^{5 x}$

Step 2

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

$\frac{d}{d x} [e^{5 x} y] = \sin (8 x)$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [e^{5 x} y] d x = \int \sin (8 x) d x$

Step 4

Integrate the left side.

$e^{5 x} y = \int \sin (8 x) d x$

Step 5

Integrate the right side.

Tap for more steps...

Step 5.1

Let $u = 8 x$ . Then $d u = 8 d x$ , so $\frac{1}{8} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 5.1.1

Let $u = 8 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 5.1.1.1

Differentiate $8 x$ .

$\frac{d}{d x} [8 x]$

Step 5.1.1.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x]$

Step 5.1.1.3

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

$8 \cdot 1$

Step 5.1.1.4

Multiply $8$ by $1$ .

$8$

Step 5.1.2

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

$e^{5 x} y = \int \sin (u) \frac{1}{8} d u$

Step 5.2

Combine $\sin (u)$ and $\frac{1}{8}$ .

$e^{5 x} y = \int \frac{\sin (u)}{8} d u$

Step 5.3

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$e^{5 x} y = \frac{1}{8} \int \sin (u) d u$

Step 5.4

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$e^{5 x} y = \frac{1}{8} (- \cos (u) + C)$

Step 5.5

Simplify.

Tap for more steps...

Step 5.5.1

Simplify.

$e^{5 x} y = \frac{1}{8} (- \cos (u)) + C$

Step 5.5.2

Combine $\frac{1}{8}$ and $\cos (u)$ .

$e^{5 x} y = - \frac{\cos (u)}{8} + C$

Step 5.6

Replace all occurrences of $u$ with $8 x$ .

$e^{5 x} y = - \frac{\cos (8 x)}{8} + C$

Step 5.7

Reorder terms.

$e^{5 x} y = - \frac{1}{8} \cos (8 x) + C$

Step 6

Divide each term in $e^{5 x} y = - \frac{1}{8} \cos (8 x) + C$ by $e^{5 x}$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $e^{5 x} y = - \frac{1}{8} \cos (8 x) + C$ by $e^{5 x}$ .

$\frac{e^{5 x} y}{e^{5 x}} = \frac{- \frac{1}{8} \cos (8 x)}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $e^{5 x}$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{e^{5 x} y}{e^{5 x}} = \frac{- \frac{1}{8} \cos (8 x)}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{1}{8} \cos (8 x)}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Simplify each term.

Tap for more steps...

Step 6.3.1.1

Combine $\cos (8 x)$ and $\frac{1}{8}$ .

$y = \frac{- \frac{\cos (8 x)}{8}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 6.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{\cos (8 x)}{8} \cdot \frac{1}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 6.3.1.3

Multiply $\frac{1}{e^{5 x}}$ by $\frac{\cos (8 x)}{8}$ .

$y = - \frac{\cos (8 x)}{e^{5 x} \cdot 8} + \frac{C}{e^{5 x}}$

Step 6.3.1.4

Move $8$ to the left of $e^{5 x}$ .

$y = - \frac{\cos (8 x)}{8 e^{5 x}} + \frac{C}{e^{5 x}}$