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To solve this second-order linear homogeneous differential equation with constant coefficients:

\[
y'' + 2y' + 17y = 0, \quad y(0) = 4, \quad y'(0) = -3
\]

we'll follow these steps:

1. **Find the characteristic equation**:
   For the equation \( y'' + 2y' + 17y = 0 \), the characteristic equation is:
   \[
   r^2 + 2r + 17 = 0
   \]

2. **Solve the characteristic equation**:
   Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = 17 \):
   \[
   r = \frac{-2 \pm \sqrt{4 - 68}}{2} = \frac{-2 \pm \sqrt{-64}}{2} = -1 \pm 4i
   \]
   The roots are \( r = -1 \pm 4i \), which are complex.

3. **Write the general solution**:
   For complex roots of the form \( r = \alpha \pm \beta i \), the general solution is:
   \[
   y(t) = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t))
   \]
   Here, \( \alpha = -1 \) and \( \beta = 4 \), so:
   \[
   y(t) = e^{-t} (C_1 \cos(4t) + C_2 \sin(4t))
   \]

4. **Use initial conditions to solve for \( C_1 \) and \( C_2 \)**:
   - **First condition** \( y(0) = 4 \):
     \[
     y(0) = e^0 (C_1 \cos(0) + C_2 \sin(0)) = C_1 = 4
     \]
     So, \( C_1 = 4 \).

   - **Second condition** \( y'(0) = -3 \):
     Differentiate \( y(t) \):
     \[
     y'(t) = e^{-t} (-C_1 \cos(4t) - C_2 \sin(4t)) + e^{-t}(-4C_1 \sin(4t) + 4C_2 \cos(4t))
     \]
     Substitute \( t = 0 \):
     \[
     y'(0) = -C_1 + 4C_2 = -3
     \]
     With \( C_1 = 4 \):
     \[
     -4 + 4C_2 = -3
     \]
     \[
     4C_2 = 1 \Rightarrow C_2 = \frac{1}{4}
     \]

5. **Write the final solution**:
   \[
   y(t) = e^{-t} \left(4 \cos(4t) + \frac{1}{4} \sin(4t)\right)
   \]

Would you like a breakdown of each step or have any questions?

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Here are some questions that expand on the topic:

1. What would the solution look like if the initial conditions were different?
2. How would the solution change if the characteristic roots were real instead of complex?
3. How does the damping effect (from \( e^{-t} \)) influence the behavior of \( y(t) \) over time?
4. How would you verify the solution by substitution back into the differential equation?
5. How does this approach generalize to higher-order differential equations?

**Tip**: When dealing with complex roots, remember that they result in oscillatory solutions with exponential decay or growth.

Solve the initial value problem: y'' + 2y' + 17y = 0, with y(0) = 4 and y'(0) = -3.

This page provides a solution to the initial value problem y'' + 2y' + 17y = 0, with initial conditions y(0) = 4 and y'(0) = -3. It includes steps to find the characteristic equation, solve for complex roots, and apply the initial conditions to find constants. Suitable for college-level students studying differential equations.

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