Find a general solution for the given differential equation with x as the independent variable.
y triple prime plus 3 y double prime minus 9 y prime plus 5 y equals 0

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.

Find a general solution for the given differential equation with x as the independent variable.

 y triple prime plus 3 y double prime minus 9 y prime plus 5 y equals 0

Discover the step-by-step solution for finding the general solution of the differential equation y''' + 3y'' - 9y' + 5y = 0. This solution includes forming and solving the characteristic equation, handling repeated roots, and using the Rational Root Theorem. Suitable for advanced high school or college-level students studying differential equations.

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