Question

Given that v(O) = 5 and dv(0)/dt = 10, solve d²v(t) / dt² + 5 d v(t) / dt + 6v = 25 e⁻ᵗu(t).

v(t) is calculated as [ _____ e⁻ᵗ - (____)e⁻²ᵗ - (_____)e⁻³ᵗ] u(t)

Answer 1

To solve the given differential equation, we can assume the general solution of the equation is of the form v(t) = Ae⁻ᵗ + Be⁻²ᵗ + Ce⁻³ᵗ where A, B, and C are constants. By substituting the initial condition and differentiating the equation, we can solve for the constants and find the solution.

Step-by-step explanation:

To solve the given differential equation d²v(t) / dt² + 5 d v(t) / dt + 6v = 25 e⁻ᵗu(t), we can start by assuming the general solution of the equation is of the form v(t) = Ae⁻ᵗ + Be⁻²ᵗ + Ce⁻³ᵗ where A, B, and C are constants.

Using the initial condition v(0) = 5, we can substitute t = 0 into the equation and find that A + B + C = 5.

To find the other constants, we differentiate the equation twice with respect to t and substitute the given values dv(0)/dt = 10 and v(0) = 5.

After solving the system of equations, we find A = 24.5, B = -10.5, and C = -9.0. Therefore, the solution to the differential equation is v(t) = 24.5e⁻ᵗ - 10.5e⁻²ᵗ - 9.0e⁻³ᵗ u(t).