Multi-Page Printing

12 1. Differential Equations and Their Solutions global existence. You will even prove something a lot stronger in a later exercise. What can be said is that for each initial condition, p, there is a unique “maximal” solution of the differential equation with that initial condition. But before discussing this, we are going to make a simplification and restrict our attention to time-independent vector fields (which we shall simply call vector fields). That may sound like a tremendous loss of generality, but in fact it is no loss of generality at all! Exercise 1–5. Let V (x, t) = (V1(x, t),... , Vn(x, t)) be a time- dependent vector field in Rn, and define an associated time inde- pendent vector field ˜ in Rn+1 by ˜ (y) = (V1(y),... , Vn(y), 1). Show that y(t) = (x(t),f(t)) is a solution of the differential equa- tion dy dt = ˜ (y) if and only if f(t) = t + c and x(t) is a solution of dx dt = V (x, t + c). Deduce that if y(t) = (x(t),f(t)) solves the IVP dy dt = ˜ (y), y(t0) = (x0,t0), then x(t) is a solution of the IVP dx dt = V (x, t), x(t0) = x0. This may look like a swindle. We don’t seem to have done much be- sides changing the name of the original time variable t to xn+1 and considering it a space variable that is, we switched to space-time notation. But the real change is in making the velocity an (n + 1)- vector too and setting the last component identically equal to one. In any case this is a true reduction of the time-dependent case to the time-independent case, and as we shall see, that is quite important, since time-independent differential equations have special properties not shared with time-dependent equations that can be used to sim- plify their study. Time-independent differential equations are usu- ally referred to as autonomous, and time-dependent ones as nonau- tonomous. Here is one of the special properties of autonomous sys- tems. 1.1.7. Proposition. If x : (a, b) → Rn is any solution of the autonomous differentiable equation dx dt = V (x) and t0 ∈ R, then y : (a + t0,b + t0) → Rn defined by y(t) = x(t − t0) is also a solution of the same equation.