URGENT! ordinary differential equations - boat example 1. The problem statement, all variables and given/known data A boat of mass m is travelling with the velocity v0. At t=0 the power is shut off. Assuming water resistance proportional to v^n, where n is a constant and v is the instantaneous velocity, find v as a function of the distance travelled. (Note that you need to consider the two cases). Please help asap! Thanks 2. Relevant equations force balance equations 3. The attempt at a solution I've gotten my ODE to be: m*dv/dt = -kv^n with the initial condition v0 at t = 0. It says find v as a function of the distance travelled. So i went on to say dv/dt= (dv/dx)*(dx/dt)= v*(dv/dx) so now x is the independent variable not t. so, m*v*(dv/dx) = -kv^n v*(dv/dx)= -(k/m)*v^n v*(dv/dx)= -av^n where a =k/m v dv = -av^n dx integral (v/(v^n)) dv = integral (-a) dx integral (v^(1-n)) dv= -ax + c (v^(2-n))/(2-n) = -ax + c (don't know if the integral on the LHS is correct?) but x=o at t=0 so v(t=0)=v(x=0)=v0 so c= (v0^(2-n))/(2-n) therefore, (v^(2-n))/(2-n) = -ax + (v0^(2-n))/(2-n) v^(2-n) = -(2-n)ax + v0^(2-n) v^(2-n) = (n-2)ax + v0^(2-n) therefore, v= [(n-2)ax + v0^(2-n)]^(1/(2-n)) but a =k/m so v(x) = [(n-2)kx/m + v0^(2-n)]^(1/(2-n)) does this seem right at all because it seems a little messy?? Also it says to note that you need to consider the two cases. What are the two cases as I've only considered one???