Question

Obtain the solution of the differential equation:

1) 2x" + 7x' + 3x = 0, X(0) = 3, x'(0) = 0
2) x" + 3x' + 6x = 0, X(0) = 3, x'(0) = 0
3) ȳ + 4ȳ + 3y = 2r(t)
where the initial are y (0) = 0, ȳ (0) = 0 and r(t) = 1, t ≥ 0
4) d²f(t)/dt² + 5 df(t)/dt + 4f(t) = e⁻ᵗu(t)
assume that all the initial conditions are zero

Answer 1

The solution to each differential equation is provided with their respective initial conditions.

Step-by-step explanation:

To solve the given differential equations:

1. For the differential equation 2x" + 7x' + 3x = 0 with initial conditions x(0) = 3 and x'(0) = 0, the solution is x(t) = 3e^(-t) - 2e^(-3t).
2. For the differential equation x" + 3x' + 6x = 0 with initial conditions x(0) = 3 and x'(0) = 0, the solution is x(t) = 3e^(-1.5t)cos(sqrt(14.75)t).
3. For the differential equation ȳ + 4ȳ + 3y = 2r(t) with initial conditions y(0) = 0 and ȳ(0) = 0 (and r(t) = 1 for t ≥ 0), the particular solution is y(t) = (1/4)e^(-t) - (1/10)e^(-3t) + (1/20)e^(-t)sin(sqrt(2)t) - (1/20)e^(-t)cos(sqrt(2)t).
4. For the differential equation d²f(t)/dt² + 5df(t)/dt + 4f(t) = e^(-t)u(t) with zero initial conditions, the particular solution is f(t) = 0.5e^(-t)u(t).