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Selesaikan soal nomor 14 secara lengkap dan detail

To solve problem 14, we start with the given Bernoulli differential equation:

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

### Step 1: Rewrite in Standard Bernoulli Form
The standard Bernoulli form is:
\[
y' + p(x)y = q(x)y^n,
\]
where \( n \neq 0, 1 \).

Rearrange the equation:
\[
y' = 4y + 2e^x \sqrt{y}.
\]
This can be written as:
\[
y' - 4y = 2e^x y^{1/2}.
\]

Here, we identify \( p(x) = -4 \), \( q(x) = 2e^x \), and \( n = \frac{1}{2} \).

### Step 2: Transform the Equation
To solve this Bernoulli equation, we use the substitution \( v = y^{1 - n} = y^{1 - 1/2} = y^{1/2} \). Thus, \( y = v^2 \).

Differentiating both sides with respect to \( x \):
\[
y' = 2v v'.
\]

Substitute \( y = v^2 \) and \( y' = 2v v' \) into the original equation:
\[
2v v' - 4v^2 = 2e^x v.
\]

### Step 3: Simplify the Equation
Divide through by \( 2v \) (assuming \( v \neq 0 \)):
\[
v' - 2v = e^x.
\]

This is now a linear differential equation in terms of \( v \) and \( x \).

### Step 4: Solve the Linear Differential Equation
The equation is:
\[
v' - 2v = e^x.
\]

This is a first-order linear differential equation, which we can solve using an integrating factor. The integrating factor \( \mu(x) \) is given by:
\[
\mu(x) = e^{\int -2 \, dx} = e^{-2x}.
\]

Multiply both sides by the integrating factor \( e^{-2x} \):
\[
e^{-2x} v' - 2e^{-2x} v = e^{-x}.
\]

The left side is now the derivative of \( v e^{-2x} \):
\[
\frac{d}{dx}(v e^{-2x}) = e^{-x}.
\]

Integrate both sides with respect to \( x \):
\[
v e^{-2x} = \int e^{-x} \, dx = -e^{-x}.
\]

Thus:
\[
v e^{-2x} = -e^{-x} + C,
\]
where \( C \) is the constant of integration.

Now, solve for \( v \):
\[
v = -e^{x} + C e^{2x}.
\]

### Step 5: Substitute Back to Find \( y \)
Recall that \( v = y^{1/2} \), so \( y = v^2 \).

Substitute \( v = -e^{x} + C e^{2x} \):
\[
y = \left(-e^{x} + C e^{2x}\right)^2.
\]

### Final Answer
\[
y = \left(-e^{x} + C e^{2x}\right)^2.
\]

This is the general solution to the differential equation.

Would you like a further breakdown of any step, or have questions on specific parts of this solution?

---

Here are five related questions that could extend your understanding:

1. How would the solution change if the power of \( y \) in the equation were different (e.g., \( y^{3/2} \))?
2. What are other common methods to solve non-linear differential equations?
3. How is the integrating factor derived and why is it effective in solving linear equations?
4. How does the value of \( n \) in a Bernoulli equation affect the substitution method?
5. Can this solution method be used for partial differential equations?

**Tip:** When solving Bernoulli equations, always check if the power \( n \) simplifies the substitution, making it easier to handle the equation in its transformed form.

Detailed solution for the Bernoulli differential equation y' - 4y = 2e^x √y, involving transformations, substitutions, and integration. This step-by-step guide is suitable for undergraduate students studying advanced calculus or differential equations.

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