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The images you've provided detail the method for forming a differential equation from a given general solution. I'll outline and summarize the key steps presented:

1. **Given general solution:**
   \[
   y(x) = e^{-2x} (A\cos 2x + B\sin 2x)
   \]

2. **First derivative of \( y(x) \) with respect to \( x \):**
   Differentiate the given function:
   \[
   y' = -2e^{-2x}(A\cos 2x + B\sin 2x) + e^{-2x}(-2A\sin 2x + 2B\cos 2x)
   \]
   Simplified form:
   \[
   y' + 2y = e^{-2x}(-2A\sin 2x + 2B\cos 2x)
   \]

3. **Second derivative of \( y(x) \):**
   Differentiate again:
   \[
   y'' + 2y' = -2e^{-2x}(-2A\sin 2x + 2B\cos 2x) - 4y
   \]
   Simplified expression for \( y'' + 2y' \):
   \[
   y'' + 2y' = -2(y' + 2y) - 4y
   \]

4. **Final form of the differential equation:**
   After solving, we derive the second-order homogeneous linear differential equation:
   \[
   y'' + 4y' + 8y = 0
   \]

This method uses differentiation and simplification to form a second-order differential equation that corresponds to the general solution provided.

Would you like further details or have any questions about this process? Here are five related questions you can explore:

1. How do you derive a particular solution from a given general solution for a differential equation?
2. Why is the homogeneous form of a differential equation important in this context?
3. Can this method be applied to other forms of solutions involving exponential and trigonometric functions?
4. How can you verify that the differential equation is correct by substitution?
5. What are the common applications of this type of second-order differential equation?

**Tip:** Always simplify derivatives carefully, as small algebraic mistakes can propagate through subsequent steps, leading to incorrect final equations.

Form a DE whose general solution is given by y(x) = e^(-2x)(Acos2x + Bsin2x)

Learn how to derive a second-order homogeneous linear differential equation from a general solution involving exponential and trigonometric functions. This guide covers key steps including differentiation and simplification to form the differential equation.

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