4. Find the general solution to the differential equation 4y'' + 20y' + 25y = 0.

 To find the general solution to the given second-order linear homogeneous differential equation, 4y'' + 20y' + 25y = 0, we can use the characteristic equation method.

Step 1: Write the differential equation in the standard form, which it already is in this case. So, we can directly proceed to the next step.

Step 2: Form the characteristic equation by replacing y'' with m^2, y' with m, and y with 1. Thus, we get a quadratic equation: 4m^2 + 20m + 25 = 0.

Step 3: Factor or use the quadratic formula to find the roots of the characteristic equation. Factoring the quadratic equation in this case gives us: (2m + 5)(2m + 5) = 0.

Step 4: Solve for the roots of the equation. The roots are m1 = m2 = -5/2, since both factors give us the same root.

Step 5: Since we have a repeated root, the general solution to the differential equation will be of the form: y = C1*e^(-5/2*t) + C2*t*e^(-5/2*t), where C1 and C2 are constants determined by initial conditions or boundary conditions, and t is the independent variable, usually time.

So, the general solution to the differential equation 4y'' + 20y' + 25y = 0 is y = C1*e^(-5/2*t) + C2*t*e^(-5/2*t).