The given differential equation is:

$y'' + 10y' + 25y = 0$

Step 1: Solve the Characteristic Equation

The characteristic equation for this second-order linear homogeneous differential equation is:

$r^2 + 10r + 25 = 0$

Factoring:

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

$(r + 5)^2 = 0$

So, the repeated root is $r = -5$.

Step 2: General Solution

Since we have a repeated root $r = -5$, the general solution is:

$y(x) = C_1 e^{-5x} + C_2 x e^{-5x}$

Answer Format

From the given answer format in the image:

$y(x) = C_1 (\text{first term}) + C_2 (\text{second term})$

Thus, the correct answer to input is:

$C_1 e^{-5x} + C_2 x e^{-5x}$

Would you like a more detailed explanation or verification of any step?

Here are five related questions for further understanding:

1. How do you solve a second-order differential equation with distinct real roots?
2. What happens when a characteristic equation has complex roots?
3. Why do we multiply by $x$ when there is a repeated root?
4. How do initial conditions affect the solution of a differential equation?
5. Can we use Laplace transforms to solve this type of equation?

Tip: Always check for repeated roots in characteristic equations, as they change the form of the general solution!