By Qingkai Kong

This article is a rigorous therapy of the elemental qualitative thought of normal differential equations, in the beginning graduate point. Designed as a versatile one-semester direction yet providing adequate fabric for 2 semesters, a brief direction covers middle subject matters reminiscent of preliminary worth difficulties, linear differential equations, Lyapunov balance, dynamical platforms and the Poincaré—Bendixson theorem, and bifurcation idea, and second-order subject matters together with oscillation idea, boundary worth difficulties, and Sturm—Liouville difficulties. The presentation is apparent and easy-to-understand, with figures and copious examples illustrating the which means of and motivation at the back of definitions, hypotheses, and normal theorems. A thoughtfully conceived number of workouts including solutions and tricks make stronger the reader's figuring out of the fabric. must haves are constrained to complicated calculus and the undemanding conception of differential equations and linear algebra, making the textual content compatible for senior undergraduates besides.

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**Extra resources for A Short Course in Ordinary Differential Equations (Universitext)**

**Example text**

H-c) for each α ∈ R. Proof. (a) Since x(t) is a solution of Eq. (H-c), x (t) exists on R. This shows that the right-hand side of Eq. (H-c) is diﬀerentiable on R, and so is the left-hand side. By diﬀerentiating both sides of Eq. (H-c) we have that (x (t)) = Ax (t). This means that x (t) is also a solution of Eq. (H-c). The general conclusion is obtained by induction. (b) This can be veriﬁed by a direct substitution. 2 that among the solutions {x(k) (t) : k ∈ N0 } and among the solutions {x(t + α) : k ∈ N0 }, respectively, there are at most n linearly independent solutions.

11. Let f (t, x) ∈ C(R × Rn , Rn ) and satisfy a local Lipschitz condition in x. Assume f (t, 0) = 0. If x(t) is a solution of the equation x = f (t, x) such that x(0) = 0, show that x(t) = 0 for any t ∈ R. 12. Consider the equation y + p(t)y + q(t)y 3 = 0 with p, q ∈ C(R, R). (a) Is it possible for the function y = et − t2 /2 − t − 1 to be a solution of this equation? (b) Is it possible for the functions y1 = t and y2 = et −1 to be solutions of this equation? Justify your answers. 13. , [f (t, x1 ) − f (t, x2 )](x1 − x2 ) ≤ 0 for all (t, x1 ), (t, x2 ) ∈ R.

A) t0 = 2, a1 = 1, a2 = −1; (b) t0 = 2, a1 = 1, a2 = 0; (c) t0 = 1, a1 = 1, a2 = −1. 10. For which values of t0 , a1 , a2 and a3 , does the IVP (y )5/3 + (y )1/3 , y(t0 ) = a1 , y (t0 ) = a2 , y (t0 ) = a3 , cos t have a solution and have a unique solution, respectively? Justify your answers. 11. Let f (t, x) ∈ C(R × Rn , Rn ) and satisfy a local Lipschitz condition in x. Assume f (t, 0) = 0. If x(t) is a solution of the equation x = f (t, x) such that x(0) = 0, show that x(t) = 0 for any t ∈ R.