Question

Please solve this in Matlab

Consider the initial value problem dx -2x+y dt x(0) m, y(0) = = n. dy = -y dt 1. Draw a direction field for the system. 2. Determine the type of the equilibrium point at the origin 3. Use dsolve to solve the IVP in terms of mand n 4. Find all straight-line solutions 5. Plot the straight-line solutions together with the solutions with initial conditions (m, n) = (2, 1), (1,-2), 2,2), (-2,0)

Answer 1

Slope Field -2 0 2

(b) The origin is a saddle point.

(c) The code is:

syms u(t) v(t) m n
ode1 = diff(u) == -2*u + 0.5*v;
ode2 = diff(v) == 0*u -v;
odes = [ode1; ode2]
cond1 = u(0) == m;
cond2 = v(0) == n;
conds = [cond1; cond2];
S = dsolve(odes,conds)
uSol(t) = S.u
vSol(t) = S.v

The solution is:

u(t)=(n*exp(-t))/2 + exp(-2*t)*(m - n/2) and v(t)= n*exp(-t).

(d) The system has two straight line solutions which are y=2x and y=0.

(e)

2 1.5 0.5 -0.5 -1 -1.5 2 2 1.5 0.5 0 -0.5 -1 -1.5

The broken red line and the green line are the two straight line solutions. Also the four solutions are also plotted and the solution for the initial point (-2,0) coincides with the solution y=0 in the green line.