Question

slope fields euler’s method and integrating factors

6. Draw the direction field of the given differential equation. Based on the direction field, determine the behavior of y as t oo. If the behavior depends on the initial value of y at t 0, describe 10 points this dependency

Answer 1

Data form the given equation is initial value of y at f = 0 y' = y(y - 3)^2........(1) in order to get equilibrium point we have to put y' = 0 i.e 0 = y(y - 3)^2 y = 0: (y - 3)^2 = 0 y = 3 y = 3 the direction field diagram of the given differential equation is as follows /r/s as t < 0 y(y - 3)^2 becomes more negative i.e get more divergence y(t) > 0 the equailibrium solution tends to wards y = 3 t tends to initially all more to words y = 3 away from y = 0 so we get two equilibrium solution At y = 3 we get asymptotic state equilibrium solution \r\n near y = 0 all more away from it so y = 0 will unstable equilibrium solution given eqution y' - y(3 - y)........(2) the critical point is o y' = 0 i.e y(3 - y) = 0 y