
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]


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Verify that the given functions are solutions to the corresponding differential equations, and derive the general solutions of these equations:

1) \( y_{1}=3 \sin 5x, \quad y_{2}=\frac{1}{5} \cos 5x ; \quad y^{\prime \prime}+25y=0 \)
2) \( y_{1}=e^{3x}, \quad y_{2}=xe^{3x} ; \quad y^{\prime \prime}-6y^{\prime}+9y=0 \)
3) \( y_{1}=e^{2x}, \quad y_{2}=e^{-3x} ; \quad y^{\prime \prime}+y^{\prime}-6y=0 \).

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