To solve the given ordinary differential equation:

$4y'' + 12y' + 9y = 0$

We will solve it step by step.

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1. Rewrite the equation in standard form:

Divide through by $4$ to simplify the coefficients:

$y'' + 3y' + \frac{9}{4}y = 0$

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2. Characteristic equation:

Assume a solution of the form $y = e^{rt}$. Substituting this into the differential equation gives the characteristic equation:

$r^2 + 3r + \frac{9}{4} = 0$

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3. Solve the characteristic equation:

The quadratic formula is given by:

$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For $a = 1$, $b = 3$, $c = \frac{9}{4}$:

$r = \frac{-3 \pm \sqrt{3^2 - 4(1)(\frac{9}{4})}}{2(1)}$

Simplify the discriminant:

$r = \frac{-3 \pm \sqrt{9 - 9}}{2} = \frac{-3 \pm \sqrt{0}}{2} = \frac{-3}{2}$

Thus, $r = -\frac{3}{2}$ is a repeated root.

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4. General solution:

For repeated roots $r_1 = r_2 = -\frac{3}{2}$, the general solution is:

$y(t) = C_1e^{-\frac{3}{2}t} + C_2te^{-\frac{3}{2}t}$

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5. Verification using computer solution:

We can verify this solution using computational tools (e.g., symbolic solvers or Python). Let me know if you'd like a code demonstration.

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Final Answer:

The general solution to the differential equation is:

$y(t) = C_1e^{-\frac{3}{2}t} + C_2te^{-\frac{3}{2}t}$

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Do you want details about computational methods or further clarification? Here are 5 follow-up questions:

1. Would you like me to explain the derivation of the characteristic equation?
2. Should I plot the solution for visual understanding?
3. Do you want the computational solution using Python or MATLAB?
4. Would you like to discuss real-world applications of this type of equation?
5. Need clarification on repeated roots and their significance in solutions?

Tip: Always check the discriminant of a characteristic equation to predict the type of solution (real distinct, repeated, or complex conjugates).