To find the general solution for the differential equation:

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

we'll proceed by finding the characteristic equation and solving for its roots.

Step 1: Form the Characteristic Equation

Assume a solution of the form $y = e^{rx}$, where $r$ is a constant. Substituting $y = e^{rx}$ into the differential equation, we get:

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

Factor out $e^{rx}$ (since it's never zero):

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

Step 2: Solve the Characteristic Equation

We need to find the roots of the cubic equation:

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

To solve this, we can try finding rational roots using the Rational Root Theorem. Testing possible roots, we find:

$r = 1 \quad \text{is a root}$

Now, we can perform polynomial long division (or use synthetic division) to factor out $(r - 1)$:

$r^3 + 3r^2 - 9r + 5 = (r - 1)(r^2 + 4r - 5)$

Factor the quadratic:

$r^2 + 4r - 5 = (r + 5)(r - 1)$

Thus, the characteristic equation can be written as:

$(r - 1)(r + 5)(r - 1) = 0$

or

$(r - 1)^2(r + 5) = 0$

Step 3: Find the General Solution

The roots of the characteristic equation are:

$r = 1 \; (\text{repeated}), \quad r = -5$

Since $r = 1$ is a repeated root, and $r = -5$ is a distinct root, the general solution is:

$y(x) = (C_1 + C_2x)e^{x} + C_3e^{-5x}$

where $C_1$, $C_2$, and $C_3$ are arbitrary constants.

Conclusion

The general solution to the differential equation is:

$y(x) = (C_1 + C_2x)e^{x} + C_3e^{-5x}$

Would you like more details on any part of the solution or have any questions?

Related Questions

1. How do you find the particular solution for a differential equation with initial conditions?
2. What is the difference between a homogeneous and a non-homogeneous differential equation?
3. How can you determine the nature of the roots (real, repeated, or complex) of a characteristic equation?
4. What are the steps to perform polynomial long division for cubic polynomials?
5. How do repeated roots affect the form of the general solution of a differential equation?

Tip:

When solving differential equations, always verify your solution by substituting it back into the original equation.