Calculus Examples

$(1 + e^{x}) \frac{d y}{d x} + e^{x} y = 0$

Step 1

Check if the left side of the equation is the result of the derivative of the term $(1 + e^{x}) y$ .

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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 $f (x) = 1 + e^{x}$ and $g (x) = y$ .

$(1 + e^{x}) \frac{d}{d x} [y] + y \frac{d}{d x} [1 + e^{x}]$

Step 1.2

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

$(1 + e^{x}) y' + y \frac{d}{d x} [1 + e^{x}]$

Step 1.3

By the Sum Rule, the derivative of $1 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [e^{x}]$ .

$(1 + e^{x}) y' + y (\frac{d}{d x} [1] + \frac{d}{d x} [e^{x}])$

Step 1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(1 + e^{x}) y' + y (0 + \frac{d}{d x} [e^{x}])$

Step 1.5

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

$(1 + e^{x}) y' + y (0 + e^{x})$

Step 1.6

Add $0$ and $e^{x}$ .

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

Step 1.7

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

$(1 + e^{x}) \frac{d y}{d x} + y e^{x}$

Step 2

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

$\frac{d}{d x} [(1 + e^{x}) y] = 0$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [(1 + e^{x}) y] d x = \int 0 d x$

Step 4

Integrate the left side.

$(1 + e^{x}) y = \int 0 d x$

Step 5

Integrate the right side.

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Step 5.1

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

$(1 + e^{x}) y = 0 + C$

Step 5.2

Add $0$ and $C$ .

$(1 + e^{x}) y = C$

Step 6

Divide each term in $(1 + e^{x}) y = C$ by $1 + e^{x}$ and simplify.

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Step 6.1

Divide each term in $(1 + e^{x}) y = C$ by $1 + e^{x}$ .

$\frac{(1 + e^{x}) y}{1 + e^{x}} = \frac{C}{1 + e^{x}}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $1 + e^{x}$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{(1 + e^{x}) y}{1 + e^{x}} = \frac{C}{1 + e^{x}}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{C}{1 + e^{x}}$