To solve the third-order linear homogeneous differential equation given by:

$y''' - 5y'' + 3y' + 9y = 0,$

we can follow these steps:

1. Form the Characteristic Equation:

The characteristic equation is obtained by assuming a solution of the form $y = e^{rt}$. Substituting this into the differential equation yields:

$r^3 e^{rt} - 5r^2 e^{rt} + 3r e^{rt} + 9e^{rt} = 0.$

Dividing through by $e^{rt}$ (which is never zero) gives the characteristic equation:

$r^3 - 5r^2 + 3r + 9 = 0.$

2. Find the Roots of the Characteristic Equation:

We need to solve the cubic equation:

$r^3 - 5r^2 + 3r + 9 = 0.$

To find the roots, we can use methods like factoring, synthetic division, or numerical approaches if necessary. A quick inspection for possible rational roots using the Rational Root Theorem suggests trying $r = 3$:

$3^3 - 5 \cdot 3^2 + 3 \cdot 3 + 9 = 27 - 45 + 9 + 9 = 0.$

Since $r = 3$ is a root, we can factor $r - 3$ out of the cubic equation.

Using synthetic division or polynomial division, we divide $r^3 - 5r^2 + 3r + 9$ by $r - 3$:

$\frac{r^3 - 5r^2 + 3r + 9}{r - 3} = r^2 - 2r - 3.$

3. Factor the Quadratic:

Now, we factor the quadratic equation $r^2 - 2r - 3 = 0$:

$r^2 - 2r - 3 = (r - 3)(r + 1) = 0.$

So, the roots of the characteristic equation are:

$r_1 = 3, \quad r_2 = 3, \quad r_3 = -1.$

4. Write the General Solution:

Since we have a repeated root $r = 3$ (with multiplicity 2) and a simple root $r = -1$, the general solution of the differential equation is:

$y(t) = (C_1 + C_2t)e^{3t} + C_3e^{-t},$

where $C_1$, $C_2$, and $C_3$ are arbitrary constants determined by initial conditions (if given).

Summary of Steps:

1. Formed the characteristic equation from the differential equation.
2. Found the roots of the characteristic equation (including using synthetic division).
3. Derived the general solution based on the nature of the roots.

Would you like more details or have any questions?

Related Questions:

1. How do you solve a second-order differential equation with complex roots?
2. What method can be used to solve a non-homogeneous differential equation of higher order?
3. How does the solution change if there are complex conjugate roots?
4. What are the methods for solving systems of linear differential equations?
5. How do initial conditions affect the constants in the general solution?

Tip:

When solving higher-order differential equations, checking for simple rational roots first can simplify the solution process significantly.