1. **Problem statement:** We have the third-order homogeneous differential equation $$\frac{d^3x}{dt^3} - 9 \frac{d^2x}{dt^2} + 26 \frac{dx}{dt} - 24x = 0$$ with initial conditions $$x(0) = 5, \quad x'(0) = 3, \quad x''(0) = 10.$$ We want to find the characteristic equation, roots, general solution, unique solution, analyze stability, and interpret long-term behavior. 2. **Characteristic equation:** Replace derivatives by powers of $r$ to get $$r^3 - 9r^2 + 26r - 24 = 0.$$ This polynomial determines the behavior of solutions. 3. **Finding roots:** Try rational roots using factors of 24: test $r=1$: $$1 - 9 + 26 - 24 = -6 \neq 0.$$ Test $r=2$: $$8 - 36 + 52 - 24 = 0,$$ so $r=2$ is a root. Divide polynomial by $(r-2)$: $$r^3 - 9r^2 + 26r - 24 = (r-2)(r^2 - 7r + 12).$$ Solve quadratic $r^2 - 7r + 12 = 0$: $$r = \frac{7 \pm \sqrt{49 - 48}}{2} = \frac{7 \pm 1}{2}.$$ So roots are $r=3$ and $r=4$. 4. **General solution:** Since roots are distinct real numbers $2, 3, 4$, the general solution is $$x(t) = C_1 e^{2t} + C_2 e^{3t} + C_3 e^{4t}.$$ 5. **Apply initial conditions:** Compute derivatives: $$x'(t) = 2C_1 e^{2t} + 3C_2 e^{3t} + 4C_3 e^{4t},$$ $$x''(t) = 4C_1 e^{2t} + 9C_2 e^{3t} + 16C_3 e^{4t}.$$ At $t=0$: $$x(0) = C_1 + C_2 + C_3 = 5,$$ $$x'(0) = 2C_1 + 3C_2 + 4C_3 = 3,$$ $$x''(0) = 4C_1 + 9C_2 + 16C_3 = 10.$$ Solve system: From first: $C_3 = 5 - C_1 - C_2$. Substitute into second: $$2C_1 + 3C_2 + 4(5 - C_1 - C_2) = 3 \Rightarrow 2C_1 + 3C_2 + 20 - 4C_1 - 4C_2 = 3,$$ $$-2C_1 - C_2 = -17 \Rightarrow 2C_1 + C_2 = 17.$$ Substitute into third: $$4C_1 + 9C_2 + 16(5 - C_1 - C_2) = 10,$$ $$4C_1 + 9C_2 + 80 - 16C_1 - 16C_2 = 10,$$ $$-12C_1 - 7C_2 = -70.$$ From $2C_1 + C_2 = 17$, express $C_2 = 17 - 2C_1$. Substitute into second: $$-12C_1 - 7(17 - 2C_1) = -70,$$ $$-12C_1 - 119 + 14C_1 = -70,$$ $$2C_1 = 49 \Rightarrow C_1 = 24.5,$$ $$C_2 = 17 - 2(24.5) = 17 - 49 = -32,$$ $$C_3 = 5 - 24.5 - (-32) = 5 - 24.5 + 32 = 12.5.$$ 6. **Unique solution:** $$x(t) = 24.5 e^{2t} - 32 e^{3t} + 12.5 e^{4t}.$$ 7. **Stability analysis:** All roots $2, 3, 4$ have positive real parts, so the equilibrium $x \equiv 0$ is unstable. Solutions grow exponentially. 8. **Pharmacological interpretation:** The drug concentration $x(t)$ increases exponentially over time, which is not physically realistic for a drug elimination model. This suggests the model or parameters may not represent a stable elimination process; in practice, drug concentration should decrease or stabilize, indicating the need to revise the model or consider additional factors.