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The differential equation you provided is:

\[
y' = e^{x - 2y}, \quad y(0) = 0
\]

This is a first-order nonlinear differential equation. To solve it, we can try to separate variables and then integrate both sides.

### Step 1: Rearrange the equation for separation of variables
We want to express the equation in a form where all terms involving \( y \) are on one side and all terms involving \( x \) are on the other side. Start by rewriting the equation:

\[
\frac{dy}{dx} = e^{x - 2y}
\]

This can be rewritten as:

\[
\frac{dy}{e^{-2y}} = e^x dx
\]

### Step 2: Integrate both sides
We now integrate both sides.

- The left-hand side requires integrating with respect to \( y \):

\[
\int e^{2y} dy = \frac{1}{2} e^{2y}
\]

- The right-hand side requires integrating with respect to \( x \):

\[
\int e^x dx = e^x
\]

Thus, the equation becomes:

\[
\frac{1}{2} e^{2y} = e^x + C
\]

### Step 3: Solve for \( y

Solve the differential equation y' = e^{x - 2y}, with the initial condition y(0) = 0.

Learn how to solve the nonlinear differential equation y' = e^{x - 2y} using separation of variables, with an initial condition of y(0) = 0. The solution involves integral calculus and exponential functions, appropriate for university-level students.

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