Calculus Examples

$\frac{d y}{d x} - y = 1$ , $y = e^{x} - 1$

Step 1

Rewrite the differential equation.

$y' - y = 1$

Step 2

Find $y'$ .

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Step 2.1

Differentiate both sides of the equation.

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

Step 2.2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 2.3

Differentiate the right side of the equation.

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Step 2.3.1

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

$\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 1]$

Step 2.3.2

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

$e^{x} + \frac{d}{d x} [- 1]$

Step 2.3.3

Differentiate using the Constant Rule.

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Step 2.3.3.1

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

$e^{x} + 0$

Step 2.3.3.2

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

$e^{x}$

Step 2.4

Reform the equation by setting the left side equal to the right side.

$y' = e^{x}$

Step 3

Substitute into the given differential equation.

$e^{x} - (e^{x} - 1) = 1$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the distributive property.

$e^{x} - e^{x} - - 1 = 1$

Step 4.1.2

Multiply $- 1$ by $- 1$ .

$e^{x} - e^{x} + 1 = 1$

Step 4.2

Subtract $e^{x}$ from $e^{x}$ .

$0 + 1 = 1$

Step 4.3

Add $0$ and $1$ .

$1 = 1$

Step 5

The given solution satisfies the given differential equation.

$y = e^{x} - 1$ is a solution to $y' - y = 1$