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To solve the differential equation in problem 14:

Given:
\[
y' - 4y = 2xe^{\sqrt{y}}
\]

We recognize that this equation does not immediately fit the standard form of a Bernoulli equation, but we can attempt a substitution to simplify it.

1. **Rewrite the equation**: Observe that \(y\) is not in a straightforward form to apply the Bernoulli method directly. Let’s rewrite it to see if any substitution might help, especially focusing on terms like \(e^{\sqrt{y}}\).

2. **Possible substitution**: Since \(e^{\sqrt{y}}\) appears, let’s try a substitution \( u = \sqrt{y} \). Then \( y = u^2 \), and differentiating both sides with respect to \(x\) gives:
   \[
   y' = 2u \cdot u'
   \]

3. **Substitute into the equation**: Substitute \( y = u^2 \) and \( y' = 2u \cdot u' \) into the differential equation:
   \[
   2u \cdot u' - 4u^2 = 2x e^u
   \]

4. **Simplify**: Divide through by \(2u\) (assuming \(u \neq 0\)) to simplify:
   \[
   u' - 2u = x e^u
   \]

This transformed equation may still require additional techniques beyond the basic Bernoulli approach, such as further substitutions or an integrating factor method.

Would you like me to continue with these steps or try a different approach? 

Here are five questions related to the steps above:

1. What other methods might be useful for solving nonlinear differential equations?
2. How can the Bernoulli equation form be adapted for different types of substitutions?
3. How do we know when a substitution will simplify a differential equation?
4. What types of differential equations can be solved with integrating factors?
5. Are there specific forms or conditions that make Bernoulli's method more applicable?

**Tip:** When dealing with differential equations that don’t fit neatly into known forms, consider using substitutions based on the terms that appear unusual, such as exponentials or roots.

Learn how to solve a nonlinear differential equation y' - 4y = 2xe^(√y) using Bernoulli's method and substitution techniques. This solution explains step-by-step transformations and approaches for tackling complex differential equations.

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