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.