Final answer:
To find the values of A and B in the general solution v(t) = Ae^(-t/τ) + B, substitute the equation into the given differential equation and solve for A and B. Then, use the initial condition v(0) = -5 to determine the exact values.
Step-by-step explanation:
To find the values of A and B in the general solution v(t) = Ae^(-t/τ) + B, we can use the initial condition v(0) = -5. Plugging in t = 0 and v = -5 into the equation, we get -5 = Ae^0 + B. Since e^0 = 1, we have -5 = A + B. We also know that A and B are constants that we need to determine.
Additionally, the given differential equation is 5(dv/dt) + 4v(t) = 6. We can substitute v(t) = Ae^(-t/τ) + B into the equation and solve for A and B.
Substituting the expression for v(t), we get:
5(d(Ae^(-t/τ)/dt) + 4(Ae^(-t/τ) + B) = 6
By differentiating, we have:
5(-A/τ)e^(-t/τ) + 4(Ae^(-t/τ) + B) = 6
Simplifying the equation, we get:
(-5A/τ)e^(-t/τ) + 4Ae^(-t/τ) + 4B = 6
Setting this equal to 0, we have:
(-5A/τ)e^(-t/τ) + 4Ae^(-t/τ) + 4B - 6 = 0
This equation holds for all t, so the coefficient of each term must be equal to 0. From here, you can solve for A and B.