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To solve this Bernoulli differential equation, we start with the given equation:

\[
y' - 4y = 2e^x \sqrt{y}.
\]

This is a Bernoulli equation, identifiable by its form \( y' + P(x)y = Q(x)y^n \), where \( n = \frac{1}{2} \) in this case.

### Steps to Solve:

1. **Rewrite the Equation**: Move all terms involving \( y \) to one side.
   \[
   y' - 4y = 2e^x y^{\frac{1}{2}}.
   \]

2. **Divide by \( y^{\frac{1}{2}} \)** (to get the equation in standard form for Bernoulli):
   \[
   y^{-\frac{1}{2}} y' - 4y^{\frac{1}{2}} = 2e^x.
   \]

3. **Substitute** \( v = y^{\frac{1}{2}} \) (so \( y = v^2 \) and \( y' = 2v v' \)):
   - Substitute \( y' = 2v v' \) into the equation:
   \[
   2v v' - 4v = 2e^x.
   \]

4. **Simplify** by dividing by 2:
   \[
   v v' - 2v = e^x.
   \]

5. **Separate Variables**:
   Rearrange to isolate \( v \) and \( v' \):
   \[
   v' - \frac{2}{v} = \frac{e^x}{v}.
   \]

Then solve by integrating both

Selesaikan persamaan Bernoulli tersebut: y' - 4y = 2e^x√y

This page provides a step-by-step solution to the Bernoulli differential equation y' - 4y = 2e^x√y. Learn how to solve using substitution and integration, covering essential calculus concepts suitable for undergraduate students.

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