1.1. First-Order ODE: Existence and Uniqueness 13 Exercise 1–6. Prove the above proposition. Consequently, when considering the IVP for an autonomous dif- ferentiable equation, we can assume that t0 = 0. For if x(t) is a solution with x(0) = p, then x(t−t0) will be a solution with x(t0) = p. 1.1.8. Remark. There is another trick that allows us to reduce the study of higher-order differential equations to the case of first- order equations. We will consider this in detail later, but here is a short preview. Consider the second-order differential equation: d2x dt2 = f(x, dx dt , t). Introduce a new variable v (the velocity) and consider the following related system of first-order equations: dx dt = v and dv dt = f(x, v, t). It is pretty obvious there is a close relation between curves x(t) satisfying x (t) = f(x(t),x (t),t) and pairs of curves x(t), v(t) satisfying x (t) = v(t) and v (t) = f(x(t),v(t),t). Exercise 1–7. Define the notion of an initial value problem for the above second-order differential equation, and write a careful state- ment of the relation between solutions of this initial value problem and the initial value problem for the related system of first-order dif- ferential equations. We will now look more closely at the uniqueness question for solu- tions of an initial value problem. The answer is summed up succinctly in the following result. 1.1.9. Maximal Solution Theorem. Let dx dt = V (x) be an au- tonomous differential equation in Rn and p any point of Rn. Among all solutions x(t) of the equation that satisfy the initial condition x(0) = p, there is a maximum one, σp, in the sense that any solution of this IVP is the restriction of σp to some interval containing zero. Exercise 1–8. If you know about connectedness, you should be able to prove this very easily. First, using the local uniqueness the- orem, show that any two solutions agree on their overlap, and then define σp to be the union of all solutions. Henceforth whenever we are considering some autonomous differ- ential equation, σp will denote this maximal solution curve with initial

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