Question

dv/dt=v+vt v(0)=2 find v(2)​

Answer 1

We have

dv/dt = 1 - 2v

=> dv/1-2v = dt

Integrating both sides

f dv/1-2v = fdt

(-1/2) ln (1-2v) = t + c

=> ln (1-2v) = -2(t+c)

=> (1-2v) = e^-2(t+c)

=> 2v = 1 - e^-2(t+c)

THEREFORE, v(t) = 1/2 {1-e^-2(t+c)}

Thus we obtain v as a function of time t. Here C is the constant of integration whose value depends on the initial condition.