Find the most general solution to the differential equation 5d2ydx2+50dydx+125y=e6x

Use the letters A
and B
to denote the constants in your solution.

The given differential equation is:

\[
5 \frac{d^2y}{dx^2} + 50 \frac{dy}{dx} + 125y = e^{6x}
\]

This is a non-homogeneous linear differential equation. The general solution to such equations can be found by solving the associated homogeneous equation first and then finding a particular solution to the non-homogeneous equation.

### Step 1: Solve the Homogeneous Equation

The homogeneous equation associated with this differential equation is:

\[
5 \frac{d^2y}{dx^2} + 50 \frac{dy}{dx} + 125y = 0
\]

First, we need to find the characteristic equation by assuming a solution of the form \( y = e^{rx} \):

\[
5r^2 + 50r + 125 = 0
\]

Divide through by 5:

\[
r^2 + 10r + 25 = 0
\]

This can be factored as:

\[
(r + 5)^2 = 0
\]

So, the roots are:

\[
r = -5 \quad (\text{double root})
\]

For a double root, the general solution to the homogeneous equation is:

\[
y_h(x) = (A + Bx)e^{-5x}
\]

where \( A \) and \( B \) are constants.

### Step 2: Find a Particular Solution

For the non-homogeneous part \( e^{6x} \), we assume a particular solution of the form:

\[
y_p(x) = Ce^{6x}
\]

where \( C \) is a constant to be determined. Substitute \( y_p(x) = Ce^{6x} \) into the original differential equation:

\[
5(36Ce^{6x}) + 50(6Ce^{6x}) + 125Ce^{6x} = e^{6x}
\]

Simplifying, we get:

\[
(180C + 300C + 125C)e^{6x} = e^{6x}
\]

\[
605C e^{6x} = e^{6x}
\]

Thus, \( C \) must satisfy:

\[
605C = 1 \quad \Rightarrow \quad C = \frac{1}{605}
\]

So, the particular solution is:

\[
y_p(x) = \frac{1}{605}e^{6x}
\]

### Step 3: Write the General Solution

The general solution is the sum of the homogeneous and particular solutions:

\[
y(x) = y_h(x) + y_p(x)
\]

\[
y(x) = (A + Bx)e^{-5x} + \frac{1}{605}e^{6x}
\]

### Final Answer:

The most general solution to the differential equation is:

\[
y(x) = (A + Bx)e^{-5x} + \frac{1}{605}e^{6x}
\]

where \( A \) and \( B \) are arbitrary constants.

Would you like more details on any of these steps or have any other questions?

### Relative Questions:
1. How do you determine the roots of the characteristic equation?
2. What happens if the roots of the characteristic equation are complex?
3. How do you find the particular solution when the non-homogeneous term is different, like a polynomial or sine/cosine function?
4. Can you explain why we chose \( y_p(x) = Ce^{6x} \) as the particular solution?
5. How would the solution change if the differential equation had different coefficients?

### Tip:
When solving a non-homogeneous differential equation, always start by solving the homogeneous part first before finding the particular solution.

Explore the general solution to the differential equation 5d^2y/dx^2 + 50dy/dx + 125y = e^6x with step-by-step explanations. Understand how to solve non-homogeneous linear differential equations using the characteristic equation method. Suitable for undergraduate-level study.

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