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The problem presented is a second-order linear homogeneous differential equation:

\[
4y'' = 3y'
\]

with initial conditions:
\[
y(0) = 4, \quad y'(0) = 12.
\]

We are required to solve this using the **superposition method**.

### Step 1: Rewrite the differential equation
Divide the entire equation by 4 to simplify:

\[
y'' = \frac{3}{4}y'.
\]

This is now a first-order differential equation in terms of \( y' \), i.e.,

\[
y'' = \frac{3}{4} y'.
\]

### Step 2: Solving the differential equation

Let \( y' = v(x) \). Then, the equation becomes:

\[
v'(x) = \frac{3}{4} v(x).
\]

This is a separable differential equation. We can solve it by separating variables:

\[
\frac{dv}{v} = \frac{3}{4} dx.
\]

Integrating both sides:

\[
\ln|v| = \frac{3}{4} x + C_1,
\]

where \( C_1 \) is a constant of integration. Exponentiating both sides:

\[
v(x) = y'(x) = C_2 e^{\frac{3}{4}x},
\]

where \( C_2 = e^{C_1} \) is a constant.

### Step 3: Integrating to find \( y(x) \)

Now we integrate \( y'(x) = C_2 e^{\frac{3}{4}x} \) to find \( y(x) \):

\[
y(x) = \int C_2 e^{\frac{3}{4}x} dx.
\]

This gives:

\[
y(x) = \frac{4}{3} C_2 e^{\frac{3}{4}x} + C_3,
\]

where \( C_3 \) is another constant of integration.

### Step 4: Apply initial conditions

We now use the initial conditions to solve for \( C_2 \) and \( C_3 \).

- \( y(0) = 4 \):

\[
4 = \frac{4}{3} C_2 + C_3.
\]

- \( y'(0) = 12 \):

\[
12 = C_2 e^{\frac{3}{4} \cdot 0} = C_2.
\]

Thus, \( C_2 = 12 \).

Now substitute \( C_2 = 12 \) into the first equation:

\[
4 = \frac{4}{3}(12) + C_3,
\]

\[
4 = 16 + C_3,
\]

\[
C_3 = 4 - 16 = -12.
\]

### Final Solution:

Therefore, the solution to the differential equation is:

\[
y(x) = \frac{4}{3}(12)e^{\frac{3}{4}x} - 12,
\]

which simplifies to:

\[
y(x) = 16 e^{\frac{3}{4}x} - 12.
\]

Would you like further clarification or details on any of these steps?

---

Here are 5 related questions you might find interesting:
1. What is the significance of the superposition method in solving differential equations?
2. How do initial conditions affect the particular solution of a differential equation?
3. What is the role of the constant of integration in solving differential equations?
4. How would the solution differ if the right-hand side of the differential equation was non-homogeneous?
5. Can this method be extended to higher-order differential equations?

**Tip:** Always check that your solution satisfies both the differential equation and the given initial conditions.

Find the exact solution of the following differential equations using the superposition method. (Note: the independent variable is x) 4y'' = 3y', y(0) = 4, y'(0) = 12.

Learn how to solve the differential equation 4y'' = 3y' with initial conditions y(0) = 4 and y'(0) = 12 using the superposition method. This solution involves applying the separation of variables and integrating with respect to the given initial conditions. Suitable for undergraduate students studying differential equations.

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