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### 1. **Bestimmen Sie die Lösung des AWP:**

\[
(2x+1)y'' - 3y = 0 \quad \text{mit} \quad y(0) = 24, \, y'(0) = 0
\]

This is a second-order linear differential equation. We'll use the method of a power series to find the solution.

#### Step 1: Power Series Ansatz
The problem suggests using a power series ansatz:

\[
g(x) = \sum_{n=0}^{\infty} a_n x^n
\]

Taking the second derivative:

\[
g'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1}, \quad g''(x) = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}
\]

Substitute these into the differential equation:

\[
(2x+1)g''(x) - 3g(x) = 0
\]

Expanding the terms:

\[
(2x+1)\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} - 3\sum_{n=0}^{\infty} a_n x^n = 0
\]

Multiplying out \( (2x+1)g''(x) \):

\[
2x\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}
\]

Combine terms by shifting indices where necessary and solve for the recurrence relation of the coefficients \( a_n \).

#### Step 2: Recurrence Relation

The resulting recurrence relation is:

\[
a_{n+2} = \frac{3a_n - 2n a_n}{(n+2)(n+1)}
\]

This gives a formula to compute all coefficients based on \( a_0 \) and \( a_1 \).

#### Step 3: Initial Conditions

Using the initial conditions \( y(0) = 24 \) and \( y'(0) = 0 \), we set:

\[
a_0 = 24, \quad a_1 = 0
\]

#### Step 4: Explicit Calculation of Coefficients

We can now calculate the first few coefficients:

\[
a_2 = \frac{3a_0 - 2 \times 0 \times a_0}{2 \times 1} = \frac{3 \times 24}{2} = 36
\]

\[
a_3 = 0 \quad (\text{since} \, a_1 = 0)
\]

\[
a_4 = \frac{3a_2 - 2 \times 2 \times a_2}{4 \times 3} = \frac{3 \times 36 - 4 \times 36}{12} = -3
\]

Continue this process to calculate \( a_5 \) and beyond.

### 2. **Fourier Series:**

The function \( g(t) = t^2 + |t| \) is given, and we are asked to compute the Fourier coefficients.

The function is \( 2\pi \)-periodic, and we have:

\[
b_k = 0, \quad \text{for all} \, k
\]

#### Step 1: Fourier Coefficient \( a_0 \)

The constant term \( a_0 \) is given by:

\[
a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} g(t) \, dt = \frac{1}{2\pi} \int_{-\pi}^{\pi} (t^2 + |t|) \, dt
\]

This integral can be split at \( t = 0 \), and the values can be computed accordingly.

#### Step 2: Fourier Coefficients \( a_k \)

The Fourier coefficients \( a_k \) are given by:

\[
a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} g(t) \cos(kt) \, dt
\]

For this particular function, you can compute the integral of \( t^2 \cos(kt) \) and \( |t| \cos(kt) \) separately.

---

Would you like more details on each step or have any specific questions about the method? Here are 5 related questions:

1. How do we handle non-homogeneous terms in differential equations using a power series?
2. Why do Fourier coefficients \( b_k \) vanish for even functions?
3. How does the choice of the initial conditions \( y(0) \) and \( y'(0) \) affect the power series solution?
4. Can we approximate the solution of a second-order ODE numerically using a power series?
5. How can symmetry properties of \( g(t) \) simplify the calculation of Fourier coefficients?

**Tip:** In many cases, calculating the first few coefficients in a power series gives a good approximation of the solution for small \( x \).

Bestimmen Sie die Lösung des AWP: (2x + 1)y'' - 3y = 0 mit y(0) = 24, y'(0) = 0. Berechnen Sie auch die Fourierkoeffizienten der Funktion g(t) = t^2 + |t|.

Learn how to solve the second-order differential equation (2x + 1)y'' - 3y = 0 using a power series method, and calculate Fourier coefficients for the periodic function g(t) = t^2 + |t|. The step-by-step solution includes the power series approach, recursion relations, and Fourier expansion.

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