
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Find the general solutions of the given linear homogeneous differential equations with constant coefficients. In cases where initial conditions are provided, find the corresponding particular solutions:

1) \( 2y^{\prime \prime} - 3y^{\prime} + y = 0 \)
2) \( 9y^{\prime \prime} + 12 y^{\prime} + 4y = 0 \), \( x = 1 \), \( y = 2 \), \( y^{\prime} = 1 \)
3) \( y^{\prime \prime} + 4y^{\prime} + 5y = 0 \)
4) \( 4y^{\prime \prime} + 25y = 0 \), \( x = \pi \), \( y = 2 \), \( y^{\prime} = 1 \)

Present the answer in LaTex format: \boxed{Your answer}
user