Whatis﻿the general solutionof﻿the third-order homogeneous differential equationy'''-4y''+8y'=0 ?

y=C1+C2e4tcos(2t)+C3e4tsin(2t)

y=C1+C2e-2tcos(2t)+C3e-2tsin(2t)

y=C1+C2e2tcos(4t)+C3e2tsin(4t)

y=C1+C2e2tcos(2t)+C3e2tsin(2t)

y=C1+C2e-4tcos(2t)+C3e-4tsin(2t)

To solve the third-order homogeneous differential equation \( y''' - 4y'' + 8y' = 0 \), we first find the characteristic equation associated with it.

1. **Characteristic Equation**: 
   \[
   r^3 - 4r^2 + 8r = 0
   \]

2. **Factoring**:
   Factor out \( r \):
   \[
   r(r^2 - 4r + 8) = 0
   \]
   This gives us one root, \( r = 0 \).

3. **Quadratic Equation**: 
   Now we solve the quadratic \( r^2 - 4r + 8 = 0 \) using the quadratic formula:
   \[
   r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 32}}{2}
   \]
   \[
   = \frac{4 \pm \sqrt{-16}}{2} = \frac{4 \pm 4i}{2} = 2 \pm 2i
   \]

4. **Roots**: 
   The roots are:
   - \( r_1 = 0 \)
   - \( r_2 = 2 + 2i \)
   - \( r_3 = 2 - 2i \)

5. **General Solution**:
   The general solution to the differential equation, using the roots, is given by:
   \[
   y(t) = C_1 + e^{2t}(C_2 \cos(2t) + C_3 \sin(2t))
   \]

Thus, the correct choice from your options is:
\[
y = C_1 + C_2 e^{2t} \cos(2t) + C_3 e^{2t} \sin(2t)
\]

### Summary
- The general solution is:
  \[
  y = C_1 + C_2 e^{2t} \cos(2t) + C_3 e^{2t} \sin(2t)
  \]

### Questions for Further Understanding
1. How do we derive the characteristic equation for higher-order differential equations?
2. What is the significance of complex roots in the context of differential equations?
3. Can you provide a specific initial condition to find particular values for \( C_1 \), \( C_2 \), and \( C_3 \)?
4. How do solutions change if the equation is non-homogeneous?
5. What techniques can be used to solve non-linear differential equations?

### Tip
When dealing with differential equations, always start by finding the characteristic equation, as it provides crucial information about the behavior of the solutions.

What is the general solution of the third-order homogeneous differential equation y''' - 4y'' + 8y' = 0?

Explore the step-by-step solution to the third-order homogeneous differential equation y''' - 4y'' + 8y' = 0. This page covers finding the characteristic equation and deriving the general solution. Ideal for advanced students in Grades 11-12.

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