Solve y' = y by Separation of Variables

Problem

Solve the differential equation

$y' = y.$

Step 1: Read the Derivative Condition

The equation says that the derivative of $y$ with respect to $x$ is equal to $y$ itself:

$\frac{dy}{dx} = y.$

In other words, the function must grow or shrink at a rate that matches its current value.

Step 2: Separate the Variables

Write the derivative as $\frac{dy}{dx}$, then move the $y$ expression with $dy$ and the $x$ expression with $dx$:

$\frac{dy}{dx} = y.$

Dividing by $y$ and multiplying by $dx$ gives

$\frac{1}{y}\,dy = dx.$

Step 3: Integrate Both Sides

Now take the antiderivative of both sides:

$\int \frac{1}{y}\,dy = \int dx.$

The antiderivative of $\frac{1}{y}$ is $\ln|y|$, and the antiderivative of $1$ is $x$, so

$\ln|y| = x + C.$

Step 4: Rewrite in Exponential Form

Use the exponential function to isolate $y$:

$|y| = e^{x+C}.$

The constant can be absorbed into one arbitrary constant, giving

$y = Ce^x.$

Step 5: Check the Solution Family

Differentiate $y = Ce^x$, where $C$ is any constant:

$y' = Ce^x.$

Since this is the same as $y$,

$y' = y.$

Therefore, the general solution is

$\boxed{y = Ce^x}.$