Find the most general solution to the differential equation 5d2ydx2+50dydx+125y=e6x5d2ydx2+50dydx+125y=e6x
Use the letters AA and BB to denote the constants in your solution.

To find the most general solution to the differential equation

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

we will first solve the homogeneous equation and then find a particular solution.

### 1. Homogeneous Equation

The homogeneous equation corresponding to the given differential equation is:

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

This is a second-order linear homogeneous differential equation. Let's solve it by finding the characteristic equation:

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

We can simplify this equation by dividing through by 5:

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

This factors as:

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

So, the roots are:

\[
r = -5, \quad -5.
\]

Since we have a repeated root, the general solution to the homogeneous equation is:

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

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

### 2. Particular Solution

To find a particular solution, \( y_p(x) \), we look for a solution of the form \( y_p(x) = Ce^{6x} \) since the right-hand side of the differential equation is \( e^{6x} \).

Substituting \( y_p(x) = Ce^{6x} \) into the original differential equation:

\[
5 \frac{d^2}{dx^2} \left( Ce^{6x} \right) + 50 \frac{d}{dx} \left( Ce^{6x} \right) + 125 Ce^{6x} = e^{6x}.
\]

Calculate the derivatives:

\[
\frac{dy_p}{dx} = 6Ce^{6x},
\]
\[
\frac{d^2y_p}{dx^2} = 36Ce^{6x}.
\]

Substituting these into the equation gives:

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

Simplify:

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

\[
605Ce^{6x} = e^{6x}.
\]

So, we have:

\[
605C = 1.
\]

Thus, \( C = \frac{1}{605} \).

Therefore, the particular solution is:

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

### 3. General Solution

The general solution to the differential equation is the sum of the homogeneous solution and the particular solution:

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

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

This is the most general solution, where \( A \) and \( B \) are arbitrary constants.

Would you like further details on this solution, or do you have any questions?

Here are some related questions to consider:
1. How would the solution change if the non-homogeneous term was different?
2. What happens if the roots of the characteristic equation are complex?
3. Can you solve a similar differential equation with different coefficients?
4. How does the method of undetermined coefficients differ from variation of parameters?
5. What is the physical significance of the constants \( A \) and \( B \) in this solution?

**Tip:** When solving differential equations, always check the form of the non-homogeneous term to determine the appropriate method for finding the particular solution.

Explore the general solution to the differential equation 5d2ydx2 + 50dydx + 125y = e6x, including the homogeneous equation, particular solution derivation, and application of characteristic equations. Suitable for college-level students and enthusiasts of differential equations.

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