Question

Let V(x) denote the number of litres of fuel left in an aircraft’s fuel tank if it has flown x km. Suppose that V(x) satisfies the following differential equation: V (x) = −aV(x) − b. Here, the fuel consumption per km is a constant b > 0. The term −aV(x), with a > 0, is due to the weight of the fuel) a) solve the equation with v(0)=vo b) how many km,x,can the plane fly if it takes off with vo litres in the tank

Answer 1

a)

$V '(x) = −aV(x) − b$

$\int \frac{dV}{aV+b}=-x+c$

$ln(aV+b)/a=-x+c$

$ln(aV_0+b)/a=c$

$ln(aV+b)/a=-x+ln(aV_0+b)/a$

$x=\frac{ln\frac{aV_0+b}{aV+b}}{a}$

b)

for $V=0$ :

$x=\frac{ln\frac{aV_0+b}{b}}{a}$