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give given part in 30 words in 1line. don give side heading as given
 

to find in 30 words in 1line. dont give side heading as to find

Formulas Used: Write out all relevant formulas that will be used to solve the problem. Ensure formulas are correct and applicable.

Concept Used: Explain the main concept or principle that applies to the problem, including any theories or laws involved. The description should be clear and concise, providing a good understanding of the concept.

Complete Calculation: Show every step of the calculation without skipping any part. Each step should include the following:

Substitution of values into the formulas
Arithmetic or algebraic operations performed
Simplification or rearrangement of equations where necessary
Explanation: After every step, include a brief explanation of what was done and why it was done. This ensures that each step is clear and logically connected to the next.

Final Answer: Provide a clear and crisp final answer, emphasizing the key result or outcome of the calculation.

Additional Requirements:

Avoid the use of pronouns and ing forms in the steps.
Structure the answer in a way that is easy to understand and follow.
Ensure accuracy in every step, calculation, and explanation.

all this in simple human english words

Given part: The differential equation is \( 4x^2y'' + y = 0 \).

To find: The solution to this second-order differential equation.

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**Formulas Used:**
1. Euler-Cauchy equation for equations of the form \( ax^2y'' + bxy' + cy = 0 \).
2. General solution of the Euler-Cauchy equation is \( y(x) = x^m \), where \( m \) is determined by solving the characteristic equation.

**Concept Used:**
This is an Euler-Cauchy equation, a type of homogeneous linear differential equation. The main idea is to transform it into a simpler equation by assuming the solution is of the form \( y(x) = x^m \) and solving for \( m \) using the characteristic equation.

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**Complete Calculation:**

1. **Substitute \( y(x) = x^m \) into the differential equation:**

   \( y'(x) = mx^{m-1} \),  
   \( y''(x) = m(m-1)x^{m-2} \).

2. **Apply these derivatives to the given equation:**

   \( 4x^2(m(m-1)x^{m-2}) + x^m = 0 \),  
   Simplifying:  
   \( 4m(m-1)x^m + x^m = 0 \).

3. **Factor out \( x^m \):**

   \( x^m(4m(m-1) + 1) = 0 \).

4. **Solve the characteristic equation:**

   \( 4m(m-1) + 1 = 0 \),  
   \( 4m^2 - 4m + 1 = 0 \).

5. **Solve the quadratic equation:**

   Use the quadratic formula:  
   \( m = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(4)(1)}}{2(4)} = \frac{4 \pm \sqrt{16 - 16}}{8} = \frac{4 \pm 0}{8} \),  
   \( m = \frac{4}{8} = \frac{1}{2} \).

6. **Write the general solution:**

   Since the characteristic equation has a double root, the general solution is:  
   \( y(x) = C_1 x^{\frac{1}{2}} + C_2 x^{\frac{1}{2}} \ln(x) \).

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**Final Answer:**
The general solution to the equation \( 4x^2y'' + y = 0 \) is \( y(x) = C_1 x^{\frac{1}{2}} + C_2 x^{\frac{1}{2}} \ln(x) \).

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Would you like any details or have any questions?  
Here are 5 related questions you might consider:

1. What happens if the characteristic equation has distinct real roots?
2. How does the Euler-Cauchy method apply to non-homogeneous equations?
3. What is the difference between general and particular solutions in differential equations?
4. How do boundary conditions affect the constants \( C_1 \) and \( C_2 \)?
5. Can the Euler-Cauchy method be applied to systems of differential equations?

**Tip:** Always check for distinct, repeated, or complex roots in characteristic equations to determine the correct form of the general solution.

Solve the second-order differential equation 4x^2 y'' + y = 0.

Learn how to solve the homogeneous second-order differential equation 4x^2 y'' + y = 0 using the Euler-Cauchy method. Detailed steps include deriving the characteristic equation and finding the general solution.

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